After Napoli's defeat to Roma earlier on Saturday, Milan were able to close the gap to Rafael Benitez's side in fifth to just six points.
Milan's season has been far from impressive, with head coach Inzaghi coming under increased pressure in recent weeks, but a second successive victory - courtesy of Alessio Cerci's first Milan goal and a late Jeremy Menez strike - gives them plenty to play for in the closing weeks of the campaign.
"Since [playing Fiorentina in] Florence [a 2-1 defeat] we have started to play the football we know," he said. "In Palermo few teams have won, we played very well.
"We conceded a goal when we were tired, but then we reacted. We were the team in the ascendancy and this is the most important thing, continuing to build good scoring chances.
"I am very satisfied. The victories boost morale and bring awareness.
"In the second half we were masters of the field, and we have suffered little. We have proven to be a united group.
"I'm happy for these two consecutive wins, and so is the president."
Meanwhile, Palermo boss Giuseppe Iachini rued missed chances for his side.
"We had a great game with a lot of scoring chances and few risks," he told Sky Sport Italia. "Unfortunately we were naive, especially on the second goal we conceded."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
