Allegri added: "When we were going into the changing rooms I mentioned Manchester United and how they had managed the last five minutes of their match.well, we have not only mismanaged five minutes, but 30 minutes!"
United rallied to take a 2-1 lead in the second half before conceding a 96th-minute equaliser at Stamford Bridge in London.
Cristiano Ronaldo managed to get on the score-sheet once again, but Juventus saw their ideal start to the Serie A season come to an end on Saturday evening.
Joao Cancelo cut inside from the left-hand side and Piatek hesitated after his shot was blocked.
Lionel Messi is next best on 389.
Ronaldo reached the landmark in the 18th minute at the Allianz Stadium, finishing from close range as Genoa goalkeeper Ionut Radu made a mess of dealing with a corner.
Krzysztof Piatek, who has been in fine form this season, almost equalised with a long-range shot in the 55th minute but Wojciech Szczesny kept it out.
Juve paid heavily for failing to kill off the encounter when Bessa headed a cross from Christian Kouame into the back of the net on 67 minutes. They are four points above second-placed Napoli - who beat Udinese 3-0 later on Saturday.
The hosts had beaten Frosinone, Empoli and above all city rivals Lazio and had been hoping to improve on their sixth place in Serie A ahead of Tuesday's home Champions League game with CSKA Moscow.
It was 2-0 11 minutes into the second half when Kevin Bonifazi headed home from a corner, and although Vanja Milinkovic-Savic's freakish dismissal with 15 minutes left gave Roma hope, they could not take advantage.