Wales midfielder Ramsey will miss the second leg of the Premier League club's UEFA Champions League play-off with the Turkish side after being shown a second yellow card for pulling Oguzhan Ozyakup's shirt 10 minutes from time.
It was the first time the midfielder had seen red in his club career and Arsenal manager Wenger felt the decision from Serbian referee Milorad Mazic was extremely harsh.
"Both yellow cards for Aaron Ramsey are unbelievable," he told ITV. "When you see the fouls they have made it's not serious. Just not serious."
Arsenal were not at their best, but almost snatched a winner when substitute Alex Oxlade-Chamberlain's late shot was tipped onto the post by Besiktas goalkeeper Tolga Zengin.
Frenchman Wenger, who also lost midfielder Mikel Arteta to injury, is optimistic that his side can progress into the group stage for a 17th consecutive season by winning the second leg at the Emirates Stadium next Wednesday.
"It was a difficult game played at a frenetic pace from the first minute to the last." he added.
"We could have won it but they had chances as well, in the first half alone. In the end we didn't lose and hopefully we will make it at home."
Wenger was also encouraged by another assured performance from recent signing Calum Chambers at the heart of his defence.
He said: "I think he was once again very good and very convincing in his performance. He coped with the pressure, won the challenges and played on a pitch that was not easy. He did very well."
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