The Nerazzurri had won just one of their previous six league games before snatching a 3-0 triumph against the Sicilian team on Sunday to climb to 10th in Serie A.
“We hope the patient (Inter) has finally healed,” Mancini told Inter.it. “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 Walter Mazzarri at the helm in November.
“It’s clear that when you win, things seem different,” Mancini said. “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 take on Celtic in Glasgow in the first leg of their Europa League last-32 clash next Thursday.
Inter are without injured defender Yuto Nagatomo, who aggravated a hamstring injury at the weekend and will be sidelined for several games.
Atalanta, meanwhile, lost 3-2 at Fiorentina last time out, their second defeat in their last three league games.
The outcome in Florence dropped the Bergamaschi to 16th place, just four points clear of the drop zone.
But Atalanta have not lost at home since November 22, while Inter have won just twice on the road this season.