The result lifted Arsenal one place up to fifth on 20 points from 11 games, behind fourth-placed West Ham on goal difference, with the Hammers at home to third-placed Liverpool this afternoon.
Smith-Rowe netted Arsenal's 100th league goal under Arteta as he drilled in a low shot from 18 metres after striker Pierre-Emerick Aubameyang had missed a first-half penalty for the home side, who also had two goals ruled out for offside.
Bukayo Saka's early effort was disallowed and Watford goalkeeper Ben Foster smothered Aubameyang's 35th-minute spot-kick before he also kept out a close-range Gabriel header with a superb reflex save.
But Foster was powerless in the 56th minute when Arsenal defender Ben White surged forward after winning possession in midfield and teed up Smith-Rowe, who found the back of the net with a first-time shot.
Aubameyang had a 73rd-minute goal chalked off and Joshua King missed Watford's solitary chance in the 86th before the visitors had Juraj Kucka sent off for a second yellow card after he fouled Arsenal's left back Nuno Tavares.
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