Spurs goalkeeper Paul Robinson saved PSV's second penalty from Danko Lazovic, a stop that provided Jenas with his chance to send Ramos's side through to the last eight.
But the England midfielder's effort was pawed away by Gomes and Chimbonda's subsequent miss ended Spurs' hopes of lifting a second cup this season after their Carling Cup success.
Gomes had an erratic game, frequently keeping his side in the game, only to then endanger his team with a rush of blood to the head.
Spurs were desperately unlucky not to win the game in the closing minutes of extra-time when another volley, this time from Steed Malbranque, was brilliantly tipped over the bar by the unpredictable Gomes.
A tense game came to life after the break when Ramos, who guided Sevilla to Uefa Cup success in 2006 and 2007, went for a broke with his substitutions, boosting his attack but at times leaving Spurs open at the back.
It made for a gripping game, even more so after a distinctly uninspiring first half - neither goalkeeper had had a save to make - as Ramos introduced striker Darren Bent, withdrawing defender Young-Pyo Lee.
Bent had a reasonable chance with his very first touch, but his shot lacked sufficient power and Gomes was able to gather the ball low to his left.
If Bent should have done better with that chance his miss was nothing compared to Danny Koevermans's woeful touch when he only had Spurs goalkeeper Paul Robinson to beat.
Soon after Edison Mendez went close with a free-kick as the game really warmed up.
Just before the hour Ramos made his second substitution, bringing on winger Aaron Lennon for Ledley King as Didier Zokora moved to the centre of the defence.
While Spurs regrouped after that second substitution, Ibrahim Afellay came close to giving PSV with the lead when his shot was beaten away by Robinson.
Jefferson Farfan was the next PSV player to trouble Robinson, this time with a flighted free-kick that the Spurs goalkeeper could only watch as it drifted narrowly over the bar.
PSV then allowed Bent to go clear, but the Spurs striker seemed unsure whether to pass or shoot and his scuffed effort neither troubled Gomes nor found Robbie Keane.
With time running out Berbatov kept Spurs' Uefa Cup hopes alive with a goal that took the breath away.
Berbatov's touch and movement - the Bulgarian was always happy to drop deep and pick up the ball - had always drawn the attention in a distinctly ordinary opening 45 minutes.
But he had had a quiet second half until he met Chimbonda's cross with an exquisite right-foot volley that flashed past Gomes.
Bent might have won the game for Spurs just before the match went to extra-time when his header was expertly tipped over by the PSV keeper.
Early in extra-time Bent almost replicated Berbatov's volley after he was sent clear by the Bulgarian.
Volleys were clearly in vogue as Jason Culina's powerfully hit effort was bravely headed clear by Jenas.
And so to penalties and misery and disappointment for Spurs. They deserved more.
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