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Andre Villas-Boas's side continued to apply pressure on their visitors and eventually equalised in the 10th minute when Michael Dawson headed home Gylfi Sigurdsson's corner past goalkeeper Adam Federici.
Jermain Defoe came close for Tottenham in the 38th minute when he got the better of Mikele Leigertwood outside the Reading penalty area, but his shot from 20 yards shaved the outside the post on its way out.
The London outfit were quicker off the blocks in the second half and went 2-1 up in the 51st minute, as Adebayor headed home Lennon's cross from close range.
Lennon himself could have added his name on the scoresheet minutes later with two quick attempts, but on both occasions he saw his efforts blocked.
Reading made a late rally for an equaliser, but their hard work was undone when Clint Dempsey scored Spurs' third goal with a shot from 25 yards that deflected off Leigertwood and into the net.