Kenneth Vargas hit a late winner as Hearts came through a difficult encounter at Cappielow to reach the Scottish Gas Scottish Cup semi-finals.
The Coast Rican forward fired home in the 86th minute to secure a 1-0 victory over Morton, who had been denied several times by some impressive Craig Gordon saves.
Hearts began to find their attacking form in the final half hour and Vargas powered home a low angled drive after Beni Baningime drove forward and fed Lawrence Shankland.
The victory seals a second semi-final for Steven Naismith’s side this season and leaves them 90 minutes away from a potential fourth Scottish Cup final in six seasons.
The match pitted two of Scotland’s recent in-form sides against each other. Morton had gone on a 16-match unbeaten run, which took them from bottom of the cinch Championship into the promotion play-off places, before suffering home defeats against Inverness and Dundee United in their previous two matches.
Although manager Dougie Imrie had lost the likes of George Oakley, Iain Wilson and Jai Quitongo since the fifth-round win over Motherwell, former Rangers defender Kirk Broadfoot returned from injury to take his place in the back three.
Hearts had won 12 of their previous 15 games and had Vargas back in their line-up after the attacker missed their victory over Celtic.
Morton signalled their intent to test Hearts in the air when Lewis Strapp hurled an early throw-in into the box from 10 yards inside the visiting half.
Another ball into the Hearts box drew the first chance as Gordon used his frame to block Darragh O’Connor’s header following an Alan Power free-kick.
The away supporters had to wait until the half-hour mark before seeing their team come close – Shankland’s first-time shot on the turn was pushed wide by Ryan Mullen.
Alan Forrest saw two efforts deflected wide and Morton were forced deep in the latter stages of the half, but Gordon had to watch Robbie Muirhead’s free-kick as it bounced into his arms.
Both sides stepped up their attacking threat at the start of the second half.
Morton had some initial pressure, but Hearts midfielder Baningime missed the chance of the game so far when he sent a free header over the crossbar from Stephen Kingsley’s cross.
The home side responded by creating two equally strong opportunities.
Gordon pulled off excellent stops from Jack Baird’s header and Strapp’s effort after the left-back had burst into the box.
Mullen was also impressing in the opposite goal and he produced a diving save from Cammy Devlin’s strike from 18 yards.
The game had the sort of edge expected of the occasion and former Morton loan star Shankland was booked for pushing Strapp to the ground in front of the “cowshed” as the home fans shouted for David Dickinson to wave a red card.
The Scotland striker was being watched by international manager Steve Clarke ahead of him naming his latest squad and he set up Vargas for an excellent chance after initially looking like he might get in behind himself.
Vargas had time to pick his spot from 10 yards, but lifted the ball on to the top of the bar and over.
Shankland then set up Aidan Denholm but the substitute shot straight at Mullen, who trapped the ball between his legs.
The skipper would have a major say in the winner though as he collected Baningime’s pass and allowed Vargas to take over.
Nuno encouraged by Forest's attitude and commitment, despite Man City defeat
Nuno Espirito Santo believes Nottingham Forest will "achieve what we want" by replicating the attitude demonstrated during their defeat by Manchester City.
Forest went down 2-0 against their title-chasing opponents at the City Ground, and remain just a point clear of the relegation zone with three games remaining.
Despite the defeat, the hosts produced a spirited performance against Pep Guardiola's side, while registering 14 shots on goal compared to 11 from the Citizens - meaning they had had more shots in both meetings this season.
Forest face already relegated Sheffield United, Chelsea and fellow strugglers Burnley in their final three games, and Nuno has seen enough to be confident of securing the Reds' Premier League status for another year.
"We played a good game, were compact, organised and had chances," he told Sky Sports. "We did a good job regarding how hard it is to play against Manchester City, and the players gave it a big go.
"We are disappointed with the result, but the encouragement comes from the way we work, how committed we are, and what we have ahead of us. This is more important.
"We stay positive, and we go for the next three games. With this attitude, we will achieve what we want."
Defender Moussa Niakhate echoes his manager's thoughts, adding: "We had three big chances in the first half, but we didn't score. That was the difference.
"Everybody is working hard. We can be proud. If we perform like this in the next three games, we'll stay up."
Al-Khaleej 0-1 Al-Nassr: Ronaldo misfires on return as Laporte nets winner
Cristiano Ronaldo returned from suspension but failed to find the net as Al-Nassr secured a 1-0 win over Al-Khaleej on Saturday, with Aymeric Laporte coming up with the decisive goal.
Ronaldo was in action for the first time since being sent off for an elbow in a 2-1 Saudi Super Cup loss to Al-Hilal earlier this month, with his initial two-match ban being reduced to one on appeal.
He missed three good chances as Al-Nassr threatened to fall further behind runaway Saudi Pro League leaders Al-Hilal, but former Manchester City defender Laporte spared his blushes by nodding Marcelo Brozovic's deep free-kick into the bottom-left corner with 68 minutes gone.
Al-Nassr remain nine points adrift of Al-Hilal at the summit in spite of their win, with Jorge Jesus' team unbeaten through 28 games this season as they close in on a record-extending 18th title.
Data Debrief: Ronaldo misfires on return
Ronaldo was presented with the Saudi Pro League's Player of the Month award for March ahead of kick-off, having scored four goals in three league appearances last month.
However, his sights were off on Saturday as he failed to net despite three of the visitors' four big chances, as defined by Opta, falling his way. He led all players on the pitch for expected goals (1.27 xG), shots (five), shots on target (three) and touches in the area (nine) without reward.
Huddersfield all but relegated from Championship while League Two play-off spots decided
Huddersfield Town saw their relegation from the Championship all but confirmed after a 1-1 draw with fellow strugglers Birmingham City on Saturday.
The Terriers avoided a defeat that would have seen them relegated on Saturday, but even a win on the final day over promotion hopefuls Ipswich Town will not be enough unless they overcome a huge goal difference of 15 goals with Plymouth Argyle.
Birmingham remain in the other relegation place and now must beat Norwich City to stand a chance of staying up with Sheffield Wednesday, Plymouth and Blackburn Rovers also still in danger.
At the other end of the table, the Canaries dropped points in a 2-2 draw with Swansea City, leaving the door open for Hull City, who face Ipswich Town later today.
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