Berba struck a super goal in Saturday's 2-0 win and Allardyce says he has everything to become an Old Trafford star.
Allardyce said: "I think Berbatov's found it a little difficult to produce his very, very best, as he showed at Tottenham after a season. This one is a big season for him. Playing for United is not an easy thing to do. There is a massive pressure and every game you have to win.
"There's no hiding here, there's no easy games and there's no doing anything other than producing your best and winning games of football.
"I think perhaps he's found, as many other players have in the past, it a little difficult to adjust to because of the pressures.
"But in this game he scored a great goal to get them on their way. He's got really outstanding ability and as this season goes on you'll find out if he's good enough - he certainly was today."
So who will win out when he comes up against Didier Drogba next Sunday?
Allardyce added: "He's a completely different player to Drogba. He's not got the same strength but he'll be judged by his goals.
"Outside the box, he's as good as anybody else in this country, probably in Europe. In terms of finding the space, his touch, his vision, is really, really good.
"Dimitar has a laid-back look but he's far from laid-back. When you ask our lads who have to chase him, they'll tell you it's far from easy."