Rafael Leao's 92nd-minute header rescued a point for Milan after Lorenzo Colombo had given Genoa the lead in the 28th minute.
After their draw, AC Milan sit in second place, three points behind Inter Milan after 18 matches, while Genoa remain in 17th, just above the relegation zone.
Allegri, speaking after the match, highlighted both the positives and areas for improvement. "We had two or three favourable situations," he said. "In the second half, we had chances. Their goalkeeper made saves and we were a bit inaccurate. The team was wonderful in trying to win the match."
He also added, "We could have done better in the first half, but when opponents mark man-to-man it can disrupt rhythm. If you don't unlock the game, you must not concede a goal."
Milan dominated large periods and registered over 30 attempts on goal, but Genoa held firm until the dramatic finale.
The match took an even more dramatic turn in the 99th minute. Milan defender Davide Bartesaghi brought down Genoa's Mikael Egill Ellertsson in the box, awarding the visitors a stoppage-time penalty after a VAR review.
Romanian substitute Nicolae Stanciu stepped up to take the kick, but ballooned the effort over the crossbar, missing the chance to snatch all three points.
Genoa manager Daniele De Rossi said he took responsibility for the miss, explaining that his regular penalty takers, Ruslan Malinovskyi and Martin, had already been substituted. "It was a tough moment, but the team gave everything," he said.
The draw keeps Milan unbeaten in Serie A in 17 games (11 wins, six draws) and keeps Genoa just three points above the relegation zone. Milan now travel to face Fiorentina on Sunday, still chasing the top spot in the title race.
Inter will be without one of their key midfielders for the Champions League clash with Arsenal after confirming Hakan Calhanoglu has suffered a calf injury that is expected to keep him sidelined for around three weeks.
Juventus manager Luciano Spalletti praised his team's effort after a 5‑0 win over Cremonese at the Allianz Stadium on Monday but insisted they are still not on the same level as Inter Milan or Napoli.
Antonio Conte was sent off after a furious reaction to a VAR penalty decision, but Napoli responded with courage and quality to earn a 2-2 draw against Inter Milan at San Siro, with Scott McTominay scoring twice to rescue a point.
Lazio have confirmed the signing of Ajax midfielder Kenneth Taylor, completing a major move for the 23-year-old as he begins a new chapter in Serie A.
