For the second time this season, administration-hit Jambos claimed a 1-0 victory over their Edinburgh rivals when Stevenson struck in style in the first half.
The win means Hearts' name will be in the draw for the semi-final, which takes place at Hampden on Thursday.
The clash at Easter Road was the third cup meeting between the sides in three consecutive seasons.
Hearts famously claimed a 5-1 win in the Scottish Cup final, before Hibs dumped their rivals out of the same competition last season.
The Leith outfit's misery was then compounded by the late dismissal of skipper James McPake.
Hearts were boosted by the news that Stevenson was passed fit to start after causing concern when he was forced out of the weekend defeat to Kilmarnock with a knee problem.
However, the home side dominated from the outset and Collins saw a header crash off the post after just five minutes.
Hearts continued to find themselves under pressure and Ryan McGivern teed up a shot for Rowan Vine, which he stabbed just wide of the upright.
Jambos goalkeeper Jamie MacDonald was then called into action to block a low drive from Robertson.
In what was proving to be a one-sided exchange in the opening stages, MacDonald came to the rescue again seconds later, this time to touch Robertson's shot onto the underside of the crossbar.
Hearts managed to survive the early onslaught and their first real chance of the game fell to Stevenson but he could only ripple the side-netting with a tame effort.
But the host were stunned when Stevenson did hit the target in the 33rd minute turning and lashing a superb, unstoppable effort into the back of the net from the edge of the box.
The goal sparked wild celebrations in the away end, before Stevenson was booked by referee Willie Collum in the aftermath.
Hibs emerged for the second half determined to haul themselves back into the tie and McGivern's throw-in found Collins who sliced the shot wide.
At the other end, Kevin McHattie attempted to double Hearts' lead but Ben Williams comfortably smothered his low drive.
The home side could have been back on level terms when Vine set up Liam Craig, who only had MacDonald to beat, but the goalkeeper pulled off an impressive point-blank save to maintain his side's slim advantage.
Hibs' hopes of a comeback were dealt a blow when McPake was shown a straight red card for a challenge on Callum Paterson just outside the box in the 82nd minute.
- Sports & Recreation