The Nerazzurri had won just one of their previous six league games heading into the encounter at the San Siro, which saw them halt a three-match losing run and lifted them to ninth in the Serie A table.
Mauro Icardi scored twice after Fredy Guarin put had the hosts in front early on, and Mancini told the club’s official website he hopes they have turned a corner.
“We hope the patient (Inter) has finally healed,” he said.. “But we must continue to train hard.
“We played a great game against Palermo and things are starting to turn for the better for us.”
Inter had lost their previous two league games, a 1-0 defeat to Torino followed by a 3-1 loss at Sassuolo, to pile the pressure on Mancini, who replaced the sacked Walter Mazzarri in November.
“It’s clear that when you win, things seem different,” the Inter coach added. “But the fact is that we had done well in other games before this one.
“The only difference is that we won against Palermo.
“I believe in recent weeks we have done better than our results suggested and I’m sorry for my players because I really have seen them work hard in training.
“We still have to improve.”
Inter are 13 points adrift of third-place Napoli, who hold the last Champions League qualifying spot.
The Nerazzurri travel to Bergamo on Sunday to face Atalanta, before taking on Celtic in Glasgow in the first leg of their Europa League last-32 clash on February 19.