And the visitors’ dominance continued after the break, when Gareth Bale and Aaron Lennon ensured Harry Redknapp’s side had five different names on the score sheet to head into the second leg at White Hart Lane on August 25 in complete control of the tie.
Hearts boss Paulo Sergio made three changes to the side which beat Aberdeen at the weekend, with Andy Webster, Ian Black and Andy Driver replacing Eggert Jonsson, Arvydas Novikovas and Scott Robinson.
Meanwhile, with a host of midfielders missing through injury, Redknapp named 21-year-old Livermore in his team, with Heurelho Gomes preferred to Brad Friedel in goal and Defoe playing as a lone striker.
Hearts lined up in a defensive 4-1-4-1 formation but it took the visitors just five minutes to make the breakthrough when Defoe and Van der Vaart exchanged passes on the edge of the box, leaving the former Real Madrid man to stumble through the Hearts defence before calmly slotting home.
Van der Vaart, who scored twice in the Champions League for Tottenham last season, grabbed the game by the scruff of its neck from there, playing an instrumental role in Spurs’ second eight minutes later.
Defoe scored just nine goals for the north London side last season but, with Tottenham’s opening game of the season against Everton postponed, the England international got off the mark at the first attempt this time out.
Lennon and Van der Vaart combined to find Defoe on the edge of the box and the 28-year-old made no mistake in dispatching past Marian Kello in typically clinical fashion.
Spurs looked like they were going to score with every attack, with their crisp, one-touch passing cutting through the Hearts defence like a knife through butter and Van der Vaart running amok.
And the visitors soon put the game out of reach after 28 minutes when Livermore fed Defoe on the edge of the box before surging forward to pick up the return ball and open his Tottenham account on just his second start.
Hearts’ first half contribution was limited to shots from distance from John Sutton and Ryan Stevenson, both of whom fired wide, while Defoe sent a volley crashing against the crossbar but was flagged for offside.
The hosts emerged from the break with renewed vigour and soon forced a trio of corners which Michael Dawson had to twice clear over his own bar.
But, for all their early second half possession, Hearts were unable to find a way back into the match, while Spurs showed their class to further extend their lead with another well-worked move on 63 minutes.
Second half substitute Tom Huddlestone’s lofted through-ball found Bale and the Welshman kept a cool head to round Kello and pass the ball into the back of the net.
Redknapp introduced tricky winger Andros Townsend after 70 minutes and the 20-year-old made his mark eight minutes later, sparking a Spurs counter-attack with a inch-perfect through-ball to find Defoe, whose cross from the left was dispatched by Lennon, who had surged forward from midfield.