The Gunners came into the match with a 2-0 advantage from the first leg and only needed a draw of any manner to progress into the next stage of the competition.
Lacazette opened the scoring in the 34th minute against the run of play, curling home a lovely free-kick from around 25 yards out after stepping up to take the set-piece himself.
Pierre-Emerick Aubameyang missed a glorious chance to make it 2-0 just before the break, failing to convert from close range after a low cross into the box from Lacazette.
The home side endured a frustrating night out, failing to find the net despite coming close on a number of occasions.
Lorenzo Insigne came close to finding an equaliser for the Serie A giants in the 60th minute when he drifted in behind Maitland-Niles on the right, and tried to lift the ball over Cech but barely got his shot off the ground and fired it straight into his arms instead.
The home side had a penalty appeal turned down four minutes later when Zielinski's low cross was blocked by a sliding Lucas Torreira inside the area but the referee could hear none of it.
The Gunners held on for a famous win on Unai Emery's 50th match in charge since replacing Arsene Wenger at the beginning of the season.
Arsenal will now face Valencia in the semi-finals.