The Gunners made the trip to the San Siro on a woeful run of form following four losses in their last five outings in all competitions to heap pressure on Arsene Wenger.
Arsenal were tipped to lose in Italy given their abject performances against Manchester City and Brighton and Hove Albion over the past fortnight or so.
German playmaker Mesut Ozil teed up Henrikh Mkhitaryan to make the breakthrough in the 22nd minute to score a vital away goal in Milan.
Ramsey, who has contended with a string of injury problems in the 2017-18 season so far, doubled the visitors’ advantage in the fourth minute of added time.
The Welsh midfielder managed to get on the end of Ozil’s pass before Ramsey’s quick feet bamboozled highly-rated Milan goalkeeper Gianluigi Donnarumma.
Former Arsenal defender Keown was full of praise for Ramsey after the Wales international helped the Gunners keep their Europa League and Champions League aspirations alive.
“Ramsey wanted to get forward at every opportunity,” Keown told BT Sport.
“He showed tidy feet for his goal.”
Arsenal can qualify for the Champions League if the Gunners win the Europa League just as Manchester United managed to do last term.
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