Jeremy Menez was the hero with two goals, while Cristian Zaccardo struck a late third to give the Rossoneri their first win of 2015.
Inzaghi headed into the game under pressure and looking to avoid a third consecutive victory that would have led to further doubts over his future and the Italian feels the pressure is evident amongst his squad.
"It was a complicated challenge," he said after the game. "We had to cope with nine or 10 absences but we deserved the win, though we need to improve our quality of performance.
"The team is in a difficult period and there is a certain fear. Individual errors have occurred because of this. We have to find serenity and key players need to lead us forward."
Milan CEO Adriano Galliani said earlier on Sunday that the club could seal a deal for Parma defender Gabriel Paletta, who was an unused substitute at San Siro.
Inzaghi said he was in the dark over any potential move but admitted he was pleased with the business Milan had done in the January window.
"I thank the board for the purchase they have made in bringing in the players I asked for. Of Paletta I know nothing. I'm happy for the arrival of [Luca] Antonelli, given the physical problems of [Mattia] De Sciglio and [Christian] Abate."
Antonelli has returned the club he began his career with on a loan deal from Genoa, while Spanish midfielder Suso has also arrived from Liverpool on a permanent deal.
Sunday's result leaves Milan in eighth place in Serie A, with league leaders Juventus up next on February 7.
