The win kept the champions in the title race following leaders AC Milan's 2-0 win at Catania on Saturday, leaving them nine points behind but with a game in hand.
Fabrizio Miccoli and Antonio Nocerino had given the visitors a 2-0 lead at the break while Javier Pastore hit the post and had a penalty saved for the Sicilians.
But Pazzini's double got Inter back on level terms before Samuel Eto'o smacked home the winner from the spot.
"I never imagined having a day like this, it really was a dream," Pazzini told SkySport.
"This team isn't satisfied, we still have the desire to win. For me it was great to play with so many great players. It was a great day, I really enjoyed it."
Palermo made a blistering start, going in front on five minutes as Mattia Cassani skipped around Davide Santon and crossed for Miccoli to tap home from close range.
Inter came storming back and Eto'o crossed into the box but Diego Milito arrived a split second too late.
Lucio headed a Miacon corner straight at goalkeeper Salvatore Sirigu while Thiago Motta blazed over.
Palermo went two up on 36 minutes as Inter again failed to deal with a Cassani cross from the right, Nocerino arriving at the back post to turn inside Maicon and beat Julio Cesar at his near post.
Inter were vulnerable on the counter attack and Santon blocked a Cassani effort before Cesar got a crucial touch to stop Nocerino rounding him.
Sirigu saved from Motta with Eto'o slotting home the rebound only to be denied by an offside flag.
It should have been three before the break but Pastore's venomous drive crashed back off the upright.
Inter were back on the attack after the break as Sirigu made a brilliant one-handed save to push away Eto'o's 20-yard free-kick.
But then debutant pair Houssine Kharja and Pazzini, both on as second half substitutes, combined to haul Inter back into the game, the Moroccan midfielder chipping into the box and the Italy forward holding off his marker before turning and shooting low past Sirigu.
Cesar kept Inter in the game on 64 minutes as he saved Pastore's penalty after Motta had cynically brought down substitute Pajtim Kasami.
Pazzini grabbed his second 17 minutes from time as he headed home Maicon's free-kick at the near post.
Moments later Ezequiel Munoz was adjudged to have brought down the striker in the box and Eto'o stepped up to sweep home the penalty.
Inter had a late escape when Federico Balzaretti's fierce goalbound volley clipped Maicon's heel and then Cesar's before flying away to safety.
Napoli remained four points behind table-topping Milan in second as Serie A top scorer Edinson Cavani smashed a hat-trick in their 4-0 thrashing of Sampdoria.
Cavani took a pass from Ezequiel Lavezzi, turned inside his marker and rifled home on 16 minutes before netting a spot-kick just before the break after he was fouled.
Marek Hamsik made the game safe with a cross shot three minutes after the break before Cavani notched his third on 57 minutes as Hamsik set him up.
Udinese increased the pressure on Juventus coach Luigi Delneri as they escaped from Turin with a 2-1 victory.
Since losing their first four matches of the season, Udinese have steadily climbed the table and now sit fifth having beaten Juve and Inter in their last two games.
Claudio Marchisio gave the hosts the lead on the hour but goals from South Americans Christian Zapata and Alexis Sanchez sealed the three points for the visitors.
Juve have now lost four out of seven games since the turn of the year.
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