After a goalless first half, Alexis Sanchez drilled home to give the visitors the lead in the 62nd minute.
Substitute Olivier Giroud fired into the roof of the net six minutes later, before Aaron Ramsey made the points secure with a deflected strike in the 74th minute.
The win lifted the Gunners back into second place and above Manchester United on goal difference heading into Tuesday’s Champions League clash with Bayern Munich at Emirates Stadium.
And Wenger was pleased with his side’s battling spirit as they held firm to eventually secure the three points.
Wenger told BBC Sport: “It was an important win for us. You could see why Watford have not conceded many goals at home. They are strong and very direct and we could not make a mistake.
“When we scored the first goal I like the fact we were relentless and continued to attack. The confidence is there and once we were 1-0 up or 2-0 up we continued to play and attack.
“Physically and mentally the game became more difficult for Watford as well and we took advantage of that.”
The Arsenal boss also heaped special praise on Sanchez, who has now scored six goals in his last three Premier League games for the north London side.
The Frenchman continued: “Alexis Sanchez is relentless going forward. He always looks dangerous and he is always dangerous. He is very important for us.”
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