The 20-year-old has emerged as one of the most exciting prospects in Italian football over the past couple of seasons but has recently been short of goals and faces a fight for a regular starting spot.
Reports have suggested that his future may lie away from Milan, with Arsenal among the clubs to have been credited with an interest, but Galliani says the player remains fully committed.
"I'd like to remind people that if we are in the Champions League, then part of the credit goes to El Shaarawy," vice-president Galliani told reporters ahead of Milan's European clash with Celtic on Wednesday.
"He is only 20 years old and flew up the ranking, so growing up that quickly is difficult. Our judgement on him is absolutely positive, he has a five-year contract and we believe in him.
"He showed how much he cares about the club colours, because over the summer he could've moved to a team that was prepared to triple his wages."
Milan will be looking to get off to a good start in their Champions League campaign at home to Celtic after being drawn in a tricky group which also contains Barcelona and Ajax.
And head coach Massimiliano Allegri has admitted their only target at this stage is to make it through to the next round.
"The aim is to overcome this group," said Allegri. "In the Champions League games are different from the Italian league.
"Barcelona are the strongest side in the group so we must get more points than Celtic and Ajax, but it won't be easy because Europe is never easy.
"I think Celtic are a strong side. Let's see what happens."