Carlos Tevez and Alvaro Morata scored either side of Marco Reus leveller for Dortmund on Tuesday to send Juve into the return fixture in three weeks' time with a 2-1 advantage.
However, Allegri believes his players had chances to make the win more comfortable in Turin.
"The result is fair, maybe we could have scored more goals," he said. "We had several chances, especially in the second half. We played a very good second half I think.
"We know they are a good team, but we did what we had to do and everything went right.
"It's a shame about the goal, but these things happen. I'm happy with the lads.
"It's not easy playing a last-16 Champions League game, and it often comes down to minor incidents in such games, but the team responded well when they saw we could attack them deep.
"I'm not worried for the second leg. In the Champions League, particularly at this stage, you're up against the best teams in Europe and you've got to be patient and organised in defence.
"This is a step forwards for a team who can do well in Europe, while remaining aware of the problems we can still have in the second leg. We're happy because we've won the home leg and above all we performed well."
Tevez's opener was cancelled out when Reus pounced on a Giorgio Chiellini slip to steal in and slot past Gianluigi Buffon, and the Argentine strike is happy with the way the hosts responded thereafter.
"They scored because we made a mistake, but we played a good match overall," he added.
"I think that the return leg will be really tough, not only because we play in their stadium, but also because in this competition you must never think that the result of a game is predictable."
Allegri also confirmed that midfielder Andrea Pirlo faces checks on Wednesday on a calf problem sustained in the game.
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