The Easter Road side claim the Edinburgh bragging rights to book a place in the fifth round of the Scottish Cup as a deflected goal seals the victory on home turf
The visitors had looked to have claimed a replay at Tynecastle next Tuesday before the midfielder's effort deflected off Marius Zaliukas to send the cup holders crashing out.
The home side stuck with the same line-up that helped them claim second place in the SPL on Wednesday when they defeated St Johnstone, while the visitors made two changes from the side who lost heavily to Celtic, with injured duo Dylan McGowan and Jason Holt replaced by Mehdi Taouil and Arvydas Novikovas.
They used the Lithuanian winger straight away, when Ryan Stevenson found Novikovas on the left-hand side and he whipped the ball in for Callum Paterson, but the young striker's effort was saved well by Hibernian keeper Ben Williams.
Leigh Griffiths had two chances from free kicks on the right wing to get the home side into the lead but, on both occasions, Jambos keeper Jamie MacDonald did well to deny the Scotland international.
Hearts almost took the lead with a helping hand from their neighbours, as Zaliukas got onto the end of a Jamie Walker corner from the right, his effort deflecting off of Paul Hanlon, but, fortunately for the Hibs skipper, Williams made a superb save to keep the sides level.
The visitors continued to pile on the pressure for the remainder of the half, with Novikovas getting the freedom of Easter Road from Alan Maybury, which allowed him to attempt a cross which cannoned off Williams' post.
Walker then had a go from the opposite wing as he crossed from the right and the goalkeeper had to be sharp to save from his own defender as Maybury slid in to clear the ball from danger.
The second half started with Darren Barr trying his luck with a couple of long-range efforts but the man who scored the opener in last season's cup final could not work the same magic this time around.
Hibs were then handed a huge blow when their top scorer, Griffiths, was stretchered from the field after picking up a knock in a clash with Barr, which meant that Pat Fenlon was forced to bring on young striker Ross Caldwell.
With just over five minutes remaining, the home side took the lead as Wotherspoon raced forward on the right hand side and unleashed a strike which smashed off of Hearts skipper Zaliukas and looped over MacDonald.
Hearts pushed forward for the closing minutes, with Zaliukas deployed as a striker along with substitute John Sutton, but the Tynecastle side could not find the equaliser and went crashing out of the Scottish Cup at the first hurdle.