As an eventful game looked set to end goalless, but Dimitar Berbatov met Pascal Chimbonda's cross with a brilliant first-time volley in the 82nd minute to leave the tie deadlocked.
They scored four against a Hammers team in freefall but, as they found at White Hart Lane, the Dutch champions were a more stubborn outfit.
It was Berbatov's moment of brilliance, in the 82nd minute that sent the game into extra-time. He met Chimbonda's driven cross with a first-time volley on the edge of the area that left Gomes with no chance.
Gomes had to tip over from Bent's header as Spurs chased a winner.
In the shootout, Jermaine Jenas and Chimbonda failed to get past PSV keeper Gomes to end coach Juande Ramos' hopes of back-to-back UEFA Cup victories this season.