The Rossoneri arrived at the Stadio Olimpico on a three-match winning run yet struggled to make an impact in the first leg of the semi-final, leaving striker Krzysztof Piatek an isolated figure up front.
Lazio made sure Tiemoue Bakayoko was shackled by consistently pressing the deep-lying midfielder when Milan had possession, though Gattuso criticised his side for not finding alternative ways to build attacks from deep.
However, the stalemate leaves Milan in a strong position to progress to the final at the expense of their opponents for a second straight year.
They needed penalties to prevail in Rome after two scoreless games between the clubs last season but have home advantage for the second leg this time around, with the fixture taking place in late April.
"It was a step back in terms of quality from recent games, but we knew it would be decided over 180 minutes," Gattuso told Rai Sport.
"We were up against a physical side in Lazio who always push four or five men up, so we did very well in defence. Unfortunately, we made a lot of mistakes in the final third and gave Piatek no service at all.
"[Joaquin] Correa was marking Bakayoko and we should have tried to play through the other midfielders, but we kept trying that central channel.
"We didn't pass it properly, we allowed five or six dangerous counter-attacks. I don't agree with those who said Lazio weren't in good shape either, as they have a way of acting sleepy and lowering the tempo, then hurting you."
Lazio had 16 shots in the contest but too often made life easy for visiting goalkeeper Gianluigi Donnarumma, managing to hit the target only four times.
Yet despite the result leaving them with work to do at San Siro, boss Simone Inzaghi was pleased with the home side's performance, particularly as they nullified opponents who have been in such excellent form.
"We were only missing a goal, but the lads put in a great performance, were always very focused against Milan, the form team in Serie A right now," Inzaghi said in his post-match interview with Rai Sport.
"We allowed Milan nothing, not even a save for a corner, but the regrets are that we didn't manage to score. At least if we were to draw at home, 0-0 was the best result."