The Gunners went in to their home leg four goals behind following lat fortnight's loss at the San Siro but came up short despite winning 3-0 to crash out of the competition 4-3 on aggregate.
Arsenal's slim hopes of a revival were heightened inside seven minutes when Laurent Koscielny, who was responsible for the own goal at Liverpool on Saturday, got free from his marker to powerfully head home Alex Oxlade-Chamberlain's firmly struck corner at the near post.
Just under 20 minutes later, the Gunners were two ahead on the night when Tomas Rosicky pounced on a poor clearance by Thiago Silva to side-foot neatly past Christian Abbiati to send the Emirates Stadium crowd into full voice.
Late in the first half, Arsenal were awarded a penalty when Djamel Mesbah crashed into Oxlade-Chamberlain just inside the edge of the area with referee Damir Skomina taking his time to point to the spot. Van Persie stepped up and hammered it into Abbiati's top left corner with the Milan gloveman going the wrong way.
The major moment of the second half came just before the hour-mark when Abbiati came up with a big save to keep out Gervinho's deflected shot before producing a tremendous stop to thwart van Persie who attempted to chip home from close range.
It was a huge effort from Arsene Wenger's side but in the end, Massimiliano Allegri's Rossoneri were able to frustrate the home side late in the piece to progress to the quarter-finals for the first time since they won the Champions League title back in 2007.
UEFA Champions League Round of 16 - second leg
Emirates Stadium, London - 59,000