The Brazilian followed up his brilliant four-goal show in the quarter-final demolition of Liverpool to rescue Arsenal after his own goal and Dimitar Berbatov's headed opener had gifted Tottenham a 2-0 halftime lead.
The former Real Madrid striker redeemed himself in stunning fashion after the break, beating England keeper Paul Robinson twice in a 12-minute spell to make Arsenal favourites to advance after next week's second leg.
Chelsea await the winners after completing a 5-1 aggregate victory over fourth division Wycombe Wanderers on Tuesday.
Manager Arsene Wenger, who did not include talisman Thierry Henry in his squad, paid tribute to Arsenal's fringe players and said he would most likely play them again in the return.
"They showed tremendous mental strength to come back," he told reporters.
"In some ways this result is more satisfying than the 6-3 victory at Liverpool because then everything went for us whereas here nothing went for us in the first half."
Tottenham, who have not beaten Arsenal in any competition since 1999, had appeared on course to end that dismal sequence.
"It's very disappointing and we only have ourselves to blame," said manager Martin Jol.
"We needed to stand up and keep the ball in the second half but we dropped off and they took control of the game. That's the law of football. You could see we were getting tired."
Bulgarian international Berbatov only managed a 15-minute cameo before succumbing to a groin injury but lasted long enough to head Jermain Defoe's cross past keeper Manuel Almunia to give Spurs a dream start after 12 minutes.
The home side doubled their advantage after 21 minutes with a goal that Baptista will want to forget.
There seemed little threat when Tom Huddlestone sent a low free kick into the area but Baptista got into a terrible tangle and dragged the ball into his own net.
Arsenal improved dramatically after the break as manager Arsene Wenger beefed up his youthful side with the introduction of Alexander Hleb and Emmanuel Eboue.
Tottenham lost their grip on the game and Arsenal cut the deficit after 65 minutes, Baptista shooting low past Robinson after the ball broke loose in the area.
As anxiety spread among the home crowd, Arsenal sensed their chance and it was no surprise when Baptista tapped in the equaliser from a low Justin Hoyte cross with 13 minutes remaining.
Defoe was then denied by Almunia in a breathless finale while teenager Theo Walcott could also have won the game for Arsenal.