Rafael van der Vaart got the Premier League side off to the best possible start when he slotted home after five minutes.
Spurs doubled their lead when Jermaine Defoe capitalised on some suspect Hearts defending.
Jake Livermore added a third before the break when he was played in by Defoe.
The hosts began the second half a lot brighter, but Harry Redknapp's side landed the killer blow when Gareth Bale rounded the Hearts keeper to make it 4-0.
Aaron Lennon sealed the rout when he finished off a counter-attacking move.
The second leg at White Hart Lane will be played on Thursday, August 25.