Catania sacked Rolando Maran for the second time this season on Monday after he had returned to replace his own successor Luigi De Canio.
Maran's latest dismissal was prompted by a 2-1 home loss to Torino that left the Sicilian side seven points adrift of safety at the foot of Serie A.
"It's too early to give up and we have to give everything until the end of the campaign," said Pellegrino, who also played for Catania.
"On Sunday we're going to San Siro to challenge AC Milan and as the coach of Catania that is a source of pride for me."
Clarence Seedorf's men are 11th in Serie A, but a run of three consecutive wins has seen them rise to within five points of city rivals Inter, who occupy fifth spot.
Despite Milan's welcome upturn in form, defender Philippe Mexes is adamant he and his team-mates will not be taking Catania lightly.
"We went through a tough period but we're getting back now and we're on the right road," Mexes told the club's official website.
"The end of the season is very important for all of us and we can't afford to underestimate Catania. There are three important points up for grabs."
