Dempsey put Tottenham ahead after a horrific error from Aleksandar Dragovic put the American clean through with Yann Sommer in no-mans land, and the former Fulham man easily rounded the keeper to slide home into an empty net.
The hosts soon came back though, with Moheme Salah toe-poking past Brad Friedel before Dragovic redeemed himself by scoring from a corner to leave Spurs in trouble. Dempsey stood up again though, getting on the end of a long-ball to prod past Sommer after the keeper made a terrible error as he somehow contrived to miss an easy save.
With time running out, though, Jan Vertonghen was given an early bath after he brought down Marco Streller when the forward was through on goal. Extra-time looms, but Tottenham will have to do it with a numerical disadvantage.