Pazzini joined Inter for euro12 million (USD 16.5 million) on Friday. He came on at halftime yesterday with Inter trailing 2-0 to goals from Fabio Miccoli and Antonio Nocerino.
He hit his first in the 57th, before Inter goalkeeper Julio Cesar saved Javier Pastore's penalty.
Pazzini scored his second in the 73rd and then earned the winning penalty that was converted by Samuel Eto'o.
"You have to say he had an excellent game," Inter coach Leonardo said. "He made his debut off the bench when we were two goals down, scored twice and won a penalty."
Napoli beat Sampdoria 4-0 at home with Edinson Cavani hitting a hattrick to stay second, four points behind league leader AC Milan on 47. Lazio is third with 40 and Inter is fourth, two points back.
"I never have and never will start thinking about where we are in the standings," Napoli coach Walter Mazzarri said. "We don't want to start thinking about that."
Udinese followed up its win over Inter a week ago by beating Juventus 2-1 at Turin's Olympic Stadium.
Following a tepid first half Juventus' Claudio Marchisio opened the scoring with an acrobatic volley after Alessandro Del Piero's shot deflected into the air.
Udinese drew level soon after when Juventus failed to clear a corner and Cristian Zapata scored from the rebound.
With five minutes remaining, Alexis Sanchez slid home Mauricio Isla's low pass to give Udinese victory.
Both teams ended the match with 10 men due to Sanchez and Juventus defender Leonardo Bonucci being sent off late on.
"We're improving and we need to keep going as we are," Zapata said. "We need to keep playing as well as we are to achieve our first aim, which is to guarantee our safety."
Also yesterday, it was; Cagliari 2, Bari 1; Genoa 3, Parma 1; Lecce 1, Cesena 1.
Palermo opened the scoring early through Mattia Cassani who took advantage of Davide Santon's error and crossed for Miccoli who had the straight forward task of slotting past Cesar.
Inter had its first chance from a Maicon corner in the 21st, but Salvatore Sirigu held Lucio's powerful header.
Nocerino then made it 2-0 when he was first to a weak punch from Cesar.
Pazzini came on at the break for Coutinho and immediately helped lift Inter's performance as Eto'o and Diego Milito forced Sirigu into saves.
Pazzini scored his first using his strength to turn in the area and shoot past Sirigu.
Palermo narrowly missed scoring a third when Thiago Motta brought down Pajtim Kasami in the area, but Pastore hit a weak penalty that Cesar saved.
The Sicilian team was left to rue the miss in the 73rd. Pazzini was left free to run on to Maicon's freekick and head his second at the near post.
Two minutes later, Inter was ahead. Ezequiel Munoz pulled down Pazzini and Eto'o converted the penalty.
"It is an excellent result," Leonardo said. "At halftime, Palermo wasn't only two goals ahead, but had a number of other goal-scoring chances. But our second half showed the force and character of this team."
Napoli opened the scoring in the 16th. Cavani took a pass from Ezequiel Lavezzi, rounded Pietro Accardi and shot past Gianluca Curci.
Just before halftime Cavani scored his second from the penalty spot, after Accardi had brought him down.
Marek Hamsik made it three for Napoli early in the second half and Cavani wrapped up his hattrick in the 57th by poking in Hamsik's cross from close range.
"Like he did against Palermo, Cavani won the match on his won," Mazzarri said.