Stanislav Lobotka's second-half strike proved enough to end a run of three straight defeats for Napoli, but it was a moment deep into stoppage time that dominated the post-match conversation.
With Lucca clean through on the counter and seconds left on the clock, the striker was dispossessed by Bologna's defence rather than simply running the ball into the corner, prompting Allegri to erupt on the touchline.
Federico Sala, a DAZN pitchside reporter at the match, described the scene to Radio Napoli Centrale.
"A classic Allegri moment - the closer it gets to the end of the game, the more he switches on and it becomes a show," Sala said. "He doesn't like it when you don't manage the ball properly. Lucca was lucky the match ended the way it did. He completely lost it, said all sorts of things. He didn't take it well. If you lose two points like that, you lose a lot."
Allegri's frustration boiled over to the point that he moved club vice-president Edoardo De Laurentiis aside as he tried to calm him down at the edge of the technical area, a moment that added further intrigue to the scenes at full time. Speaking afterwards, Allegri explained the message he had tried to get across to Lucca before sending him on.
"He looked like he was strapped to a coat hanger, and I told him to open his shoulders up," Allegri said. "These are good players who can improve. On a mental level, they need confidence."
Sala also revealed a tactical instruction picked up from Allegri during the half-time interval that shaped Napoli's second-half improvement.
"At half-time we caught a phrase from Allegri in the tunnel: he was asking the team to spread the play and go inside," Sala said. "The solution had to come through the wide areas. Lang had a great second half, he shifted the momentum of the game. He forced Bologna to sit deeper. Allegri's instructions worked."
The win eased the pressure on Allegri following back-to-back defeats to Como and Inter as well as a Champions League loss to Arsenal, though his animated reaction at the final whistle suggested the manner of victory mattered as much to him as the result itself.
Bologna, meanwhile, remain without a win through four league matches under Domenico Tedesco, with pressure growing on the club's own head coach after a disappointing start to the season.