Danny Welbeck broke the deadlock for the visitors in the seventh minute at Goodison Park as Arsenal sought to win their first Premier League game since 14 February.
Alex Iwobi, who was making his Premier League debut against the Toffees, doubled the away side’s lead with three minutes remaining of the first half.
Arsenal temporarily moved to within eight points of leaders Leicester City before the Foxes were 1-0 winners against Alan Pardew’s struggling Crystal Palace side later on Saturday.
Former Gunners defender Keown, who was part of Arsene Wenger’s last title-winning side, backed Arsenal to consolidate their top-four spot but ruled out the north Londoners winning a first Premier League crown since 2004.
“That is a great result for the Gunners’ Champions League aspirations, but I can’t see them winning the title. And that is such a disappointment, because this season it was there for the taking,” Keown told BBC Sport.
Arsenal are in action against Watford in the London derby at the Emirates Stadium on 2 April.
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