The edged Sporting Lisbon 0-1 on Thursday in the UEFA Europa League
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The striker pounced on a Sebastian Coates error and slotted a shot through the keeper's legs after 77 minutes.
Earlier in the second half, Welbeck had a headed goal ruled out after he was judged to have fouled the defender in the build-up.
Arsenal have now extended their winning run to 11 games in all competitions.
This was Arsenal's first away win in Portugal after previously failing to come away victorious in six European visits there (D3, L3).