Mark Hateley took his tally to 26 for the second goal, and there was little a severely limited Hearts could do to prevent Rangers from securing a semi- final against Kilmarnock.
Kilmarnock reached this stage of the cup for the first time in 22 years by beating Dundee 1-0 courtesy of a Tom Brown goal at Rugby Park, but the other semi-final is not so clear cut.
Dundee United were held to a scoreless draw by Airdrie at Broomfield, and the winners of Tuesday night's replay will meet either St Johnstone or Aberdeen. These two also failed to score yesterday and will come together again on Tuesday night at Pittodrie.
The semi-finals will be played on 9 April and the venues will be decided at a Scottish FA council meeting tomorrow.
Although Rangers would rather have seen Aberdeen jettisoned from the competition they remain confident of becoming the first side to win back-to-back trebles.
Their sense of well-being was reflected in the impressive manner in which Brown, normally content to patrol the back line, ambled forward as though on the training pitch to score a remarkable goal four minutes into the second half.
Hearts cleared the ball only as far as Brown, who arrived 25 yards out on the left-hand side of the pitch. He took one touch before clipping his shot towards the right-hand corner of the Hearts goal. The ball never rose more than a foot above the ground and was in the net before Henry Smith had completed his dive.
Gordon Durie, who had a productive afternoon playing wide of Hateley and Ally McCoist, found Dave McPherson and his deep cross dropped for Hateley to nod in. It was all over at that point, although there was to be one other moment of celebration for Rangers supporters, who made up the bulk of the 41,666 crowd.
Nine minutes from time Durie was withdrawn allowing space for Rangers' pounds 4m striker Duncan Ferguson. It was a rare sighting of the precocious youngster, but also another indication of the wealth of talent the manager, Walter Smith, has at his command.
His team have a formidable look about them. Their captain, Richard Gough, says it is an inner belief while McCoist claims it is confidence, but whatever makes them produce at crucial moments it is certainly a powerful force.
Kilmarnock will not wish to dwell on the fact that it has been six years since Rangers last lost in the Scottish Cup to any team other than Celtic.
TENNENTS SCOTTISH CUP Semi-final draw: Dundee United or Airdrie v St Johnstone or Aberdeen; Kilmarnock v Rangers. To be played 9 April, venues to be announced.