An Emilio Izaguirre own-goal and a Sulley Muntari effort, both in the last 10 minutes, ensured Allegri's men got their European campaign off to a winning start.
However, Allegri was pleased with the nature of his side's win, praising his charges' mental toughness to grind out the win.
"The lads have heart and desire, as they know a game is always open until the final whistle, as we saw against Torino," Allegri told Sky Sport Italia.
"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.
"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."
Milan have won just once in Serie A this season and were very nearly held at San Siro on Wednesday.
Allegri believes his side need to improve and he riticised their attempts to try and match Celtic's direct style early on.
"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," he said.
"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.
"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."
Milan midfielder Nigel de Jong felt his team-mates were lucky to get the win, claiming Milan were not in control for the first half.
"It was a lucky win but sometimes you need some luck," he said.
"I do not think we started off very well. We let Celtic dictate in the first half."
