Napoli ended a run of three straight defeats with a hard-fought victory at the Maradona, Stanislav Lobotka's 66th-minute strike settling the contest after being teed up by substitute Costantino Favasuli.
Speaking to Sky after the match, Allegri admitted his side could have managed certain moments of the game better, despite claiming a first home win of the season.
"Because in the end we could have managed some situations better, we were keeping possession in our own half when instead we need to do it in the opponent's box. We conceded very little in any case, let's enjoy this win, which is our first at home. Hopefully it's a good start."
The Napoli coach also discussed Kevin De Bruyne's role further forward in the team, closer to the striker.
"I think Kevin has played there his whole life. He's so good at creating space for himself, at understanding when he needs to drop deeper, that it's practically impossible to argue with. The way he controls the ball and the way he passes it shows he's on a completely different level. Given his age, he needs to be managed carefully."
Allegri explained that De Bruyne's late withdrawal was down to cramp rather than any injury concern, and praised the impact of substitutes Favasuli and Noa Lang.
"He gave a lot, it was cramp. Today the match was won with anger and desire, which was missing between Como and Inter. Politano worked hard in the first 45 minutes, but then we needed fresh energy. At this early stage of the season, with these temperatures, making changes at half-time every so often can help."
Bologna were without suspended head coach Domenico Tedesco, who was serving the second of a two-match ban for his conduct towards a referee, with assistant Andrea Tarozzi taking the dugout in his place.
Tarozzi, who once played under Allegri at Sassuolo, said Bologna's problems were more about a lack of cohesion than any lack of application.
"The road is long, we have to find the balance between the various parts of the team that we're currently struggling to find. We're on the right path because the principles are gradually starting to show," Tarozzi said.
Asked whether the defeat owed more to Bologna's own shortcomings or Napoli's second-half improvement, the assistant coach was candid.
"We share some of the blame too. We were organised defensively, but we can't create so little, otherwise opponents start to shut us down and it becomes a struggle."
Tarozzi also explained his decision to substitute forward Roberto Piccoli.
"He had a good game, but also an incredible workload against Napoli's two centre-backs. We wanted to try something different. We have to put aside the disappointment of the moment, it's a period where things aren't going our way at all."
The result leaves Bologna without a win after four matches, with just a single point to their name so far this season.