Substitute Pintado equalised on 63 minutes, but then turned villain in injury time when Matt Duke saved his penalty.
But fellow countryman Gomez made no mistake with Swansea's second spot-kick in extra-time after Febian Brandy was fouled.
Both sides made wholesale changes for this second round clash at the Liberty Stadium.
Tigers boss Phil Brown named an entire second string after an unbeaten start to life in the top-flight, while Swansea also fielded an under-strength line-up.
Hull went ahead after 11 minutes when veteran striker Windass showed his vast experience with a clinical left-foot finish. Nicky Barmby was the architect, seizing on a dreadful clearance by Gary Monk to set up the opener. The ex-England midfielder's through ball was dummied by striker Caleb Folan and Windass beat Dorus de Vries from 12 yards.
Hull were sharper to the ball and Ryan France had the chance to double Hull's advantage six minutes later but his header flashed wide.
There were chances at either end with Windass and Gomez failing to capitalise.
But Hull were the more threatening with strike duo Folan and Windass always a threat and should have added a second five minutes before the break when teenage defender Liam Cooper headed Bryan Hughes's corner inches wide.
Swansea responded in the second period after manager Roberto Martinez threw caution to the wind with a double substitution in the 54th minute.
The hosts spurned a great chance after Pintado skipped past Wayne Brown and cut-back for striker Guillem Bauza who blazed over from eight yards.
Swansea piled on the pressure when Bauza and then Pintado were only denied by fine saves from Duke.
But they equalised on 63 minutes when Pintado's flick header gave Duke no chance.
Pintado should have won the match in the last minute after substitute Brandy was fouled by Wayne Brown inside the box. But Duke guessed the right way and palmed away his effort before Gomez sent Swansea through in the 105th minute.