They scrambled through 4-3 on aggregate following a fraught night in north London, losing the second leg 3-0 after a woeful first half in which the hosts scored three times.
While it was mission accomplished, Milan can ill afford another 45 minutes like that if they are to have any hope of winning Europe's top club prize for an eighth time.
"The important thing is that we are through but there is nothing to celebrate," said French defender Philippe Mexes.
"In the second half we played better but in the first we had too much fear. Now we go on breathing a sigh of relief.
"The good thing was we had the character not to concede in the second half but it was a night to forget."
Defender Antonio Nocerino, who was at fault for conceding the penalty that Robin van Persie scored to make it 3-0, said his side could not afford a repeat of the first half at Emirates Stadium.
"We didn't play a good first half and we conceded three goals and risked even more," he told Milan's website. "But in the second half we played well and now we're through to the quarter finals so we can enjoy that.
"At half time we said we had to stay calm and that's what we did. I missed a sitter right in front of the goal.
"In the quarter-finals, they're all big teams but the sure thing is that they'll be up against a Milan side that'll be even better and a first half like you saw tonight will never happen again."
Milan coach Massimiliano Allegri picked an attacking line-up to defend their 4-0 advantage from the San Siro with Zlatan Ibrahimovic, Stephan El Shaarawy and Robinho deployed in a three-man attack, but it almost backfired.
The Serie A side looked ponderous and aimless in the opening period, handing the initiative to Arsenal who tore into the visitors with relish.
After the break they finally began to get the ball to their front men and Milan missed numerous chances to kill off the home side and avoid a nerve-jangling finale.
Allegri said it satisfying to be back in the last eight again.
"After four years of going out at the last 16 of the competition, I think that this evening, making it to the quarter-finals was important for the club," he said.
"We're back among the best eight in Europe and next season we'll have a better seeding as well.
"Real Madrid and Barcelona at the moment are the best teams in Europe but this team when we're at full strength we can have our say as well."
