The defending champions suffered their first loss of the season - and second in 2014 - as Gianluigi Buffon's 500th Serie A appearance ended on a sour note.
Juve - who saw Fernando Llorente and Angelo Ogbonna denied by the woodwork - conceded in the 94th minute with Luca Antonini prodding home from six yards to end a swift counter-attack.
Allegri told Sky Sport Italia: "We played on a simply awful pitch. You cannot play football on that field.
"I have said before that Scudetti [Serie A titles] are also earned with draws. We fought hard and tried everything to win, but we can't concede right at the death.
"To win the title you don't need 108 points, but just one more than the team in second place.
"Do everything to win, but if you hit the woodwork, see some great saves and are 30 seconds from the end, try not to lose.
"We leave here with regrets, as we were six against two on that counter and conceded a goal. That has to make us think and improve in terms of maturity."
Allegri rebuffed suggestions that he should be concerned about the club's recent away record, though.
Juve have not won in their last four road trips, with Champions League defeats at Atletico Madrid and Olympiacos sandwiched by a 1-1 league draw at Sassuolo.
"I am not worried, despite the statistics," he added.
"Against Sassuolo we did everything possible and a draw can happen, [on Wednesday] we did everything and conceded right at the end.
"Just as it was an important point against Sassuolo, we should've picked up an important point here.
"When Genoa had the ball, we should've focused on defending, not trying to create another chance to go forward.
"I hope this can serve as a lesson for the future, as it's practically impossible to win every game, so every point counts."
