Following Fabian Johnson's opening goal for the hosts, Pogba delivered a spectacular, perfectly-weighted lob to set Stephan Lichtsteiner up for an equalising goal that he struck with relish, having just returned to the Juventus side after heart surgery.
A win would have seen Allegri's side safely through to the Champions League knockout stages, but the sending off of Hernanes for a dangerous challenge on Alvaro Dominguez hampered Juve's chances of scoring a late winner.
Pogba's performance belied his struggle for form in the early part of the season, and Allegri praised the France midfielder but also called for him to improve.
He said: "He’s growing. Tonight, he put in an important performance but he still has to improve.
"He needs to read certain moments of the game better.”
Victory over Manchester City in their next Champions League Group D game will be enough to send Juventus into the last-16, and Allegri wants his side to show more character.
He said: "Our character as a team must grow but we must also improve in a technical sense.
"We shouldn’t struggle on the sign of pressure from the opposition. If we are not able to keep the ball, we should play it long into our strikers, at the very least.
"I think that the team still has to grow and stop being overcome by nerves. There’s only one ball and we have to play well. We have to read certain situations better."
Monchengladbach had been in free-scoring form in the run up to the game, and midfielder Ibrahima Traore lamented their failure to continue their goal glut against Juventus, saying: "We had so many good chances, but if you don't make them count, you cannot win a match like this."