The 21-year-old Ghana international featured only twice for the Gunners this season - both in the Capital Cup - and had been on loan at Fulham since January.
The Arsenal club website said: "We wish Manny the best of luck in his recovery."
Frimpong, who also had a six-week loan spell at Championship side Charlton Athletic last year, picked up the injury on his right knee during Fulham's Premier League match at Swansea City on Sunday.
He had his operation on Wednesday night and will spend the summer undergoing rehabilitation.
Meanwhile, Jack Wilshere has expressed his frustration at Arsenal's failure to win more games as they continue to trail in the wake of their rivals.
Arsene Wenger's team left it to the last day of the campaign to secure Champions League qualification and despite finishing the run-in impressively with 26 points from a possible 30, Wilshere knows the team need to step up their efforts.
"The games earlier on don't decide our season but we have got to think that we have to win them, because that is what champions do," he told the club's official website.
"The likes of Manchester United go away and win games and leave you thinking, 'they keep winning and winning' and at the end of the season the points tally shows that.
"That is what we have to do [to challenge for the title], and we also have to improve our record against the big teams. It has not been great this season and if you want to become champions you have to beat those sides."