The Wales international pulled up in the second half with a suspected muscular tear and was replaced by Alex Oxlade-Chamberlain in the 57th minute at Emirates Stadium.
And Wenger has revealed that the 24-year-old is set for a spell on the sidelines after suffering the problem.
Wenger told his post-match news conference: “He [Ramsey] is alright but very down because he told me that his hamstring is quite a tough one. He is out.”
The Arsenal boss also admitted that Alexis Sanchez had been struggling with fatigue during the game after a busy few days for Arsenal’s talismanic Chile international.
The Gunners manager continued: “Alexis, you could see he was a bit jaded physically tonight and he still pushes himself so much but he finished exhausted.
“The two games with Chile, plus travelling, plus Watford, plus tonight, it is too much.”
After a goalless first half, substitute Olivier Giroud bundled home in the 77th minute after goalkeeper Manuel Neuer missed Santi Cazorla’s free-kick.
And Mesut Ozil secured the win when the Germany international turned in Hector Bellerin’s cross in stoppage time.
Arsenal host Everton on Saturday evening.
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