A frustrating hour saw Juve waste a host of chances in Turin before they finally broke the deadlock through Carlos Tevez.
The Argentine striker ended his 14-game Champions League goal drought after playing a one-two with Kwadwo Asamoah before firing past Robin Olsen.
Tevez added his second in the closing minutes with a curling free-kick from the edge of the penalty area to seal the points in the Group A clash.
Allegri praised his side's patience to eventually break down their Swedish opponents but admitted they had highlighted areas that need to be worked on in the coming games if they wanted to progress.
"I did expect this kind of game, there's nothing easy in Europe," the head coach told reporters. "They [Malmo] defended very well, we knew we had to do well.
"We didn't play the way I wanted in the first half, but we played better after the break. We could have scored even more than two goals. It will be a complicated group, we have to improve.
"We didn't let them have many chances and that is a good thing, but we have to manage the game and the ball better.
"I am not saying that my players didn't do well, but I am sure they can do better. In any case, I am happy with the result."
Juventus' win sees them sit top of the group on goal difference after the opening matchday, with Olympiacos second after their 3-2 win over Atletico Madrid, who Juve face next.
