The result ended a run of four Premier League away loses.
Wenger made six changes in his team that lost at Crystal Palace last Monday, bringing in Laurent Koscielny, Gabriel, Rob Holding, Petr Cech, Alex Oxlade-Chamberlain and Olivier Giroud to the starting line-up.
Alexis Sanchez’s superb free-kick gave Arsenal the lead just before the break.
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Alvaro Negredo cancelled out Sanchez’s opener in the 50th minute as he volleyed in Stewart Downing’s cross.
However, Mesut Ozil secured the narrow win for the Gunners in the 71st minute from close range after a through ball from Aaron Ramsey.
The win moves the Gunners up to sixth, seven points behind fourth-place Manchester City and with a game in hand.
Middlesbrough remain deep in the drop zone, six points from safety.