The Rossoneri coach insisted he was not bothered by another late show from his players and praised them considering the squad's current injury crisis
Cristian Zapata and Sulley Muntari scored the goals in the final 10 minutes to ensure that they claimed all three points at San Siro in their opening Group H game.
Milan also left it late to secure a draw against Torino on Saturday and, although the win left home fans biting their finger nails, Allegri is pleased considering the Rossoneri's current injury woes.
"It was important to win and we struggled at the start of the second half, as we couldn’t get it clear. We were fortunate with the goal," the 46-year-old told Sky Sport.
"It was a good victory for a group of players who were whittled down to the bone, but played with substance. What counted was getting the points.
"I am not angry at always getting the goals in the final stages, as if all games went like this I’d be happy! The truth is towards the end the spaces open up more and we can create more chances.
"We do have to improve, as we gave the ball away too much and tried to force the passes through, but it was a good reaction from the side to earn victory.
"The lads have heart and desire, as they know a game is always open until the final whistle, as we saw against Torino. What we do need to improve is how we read the game, knowing when to take up the tempo or slow it down. We started playing Celtic’s game with long balls, which is what we should not be doing.
"We must always try to think positive. I have faith in the last 15-20 minutes of a game, as I know anything can happen there. The substitutions helped, as they did on Saturday."
Allegri also looked ahead to Milan's clash with fellow Serie A giants Napoli this weekend and the San Siro trainer had nothing but praise for the squad Rafa Benitez has assembled at the Stadio San Paolo.
"This result also means we can prepare as best we can for the game with Napoli on Sunday," he added. I have to compliment Rafa Benitez, who in a few months has been able to transform their playing style.
"M’Baye Niang, Ignazio Abate and Andrea Poli should all be back on Sunday, we have a few days to recover enthusiasm and fitness levels for the first big head-to-head of our season.
"I do not envy Benitez’s squad at all. I am happy with the side at my disposal, I think we’ll have a good season and in Serie A have plenty of time to make up for the early setbacks."