The French boss has come under growing scrutiny after the Gunners were beaten 3-1 by Bayern Munich at the Emirates Stadium on Tuesday in the first leg of their last-16 meeting.
Despite the disadvantage the North Londoners possess going into the return leg, Ramsey believes that his side can turn the tie around.
"Teams have been in this position before and managed to pull it off. Hopefully, we can have that sort of mentality," Ramsey told talkSPORT. "We need a bit of luck on the night but it's been done before.
"We want to do well as a team and do well for him. He's put all his faith into us and we have to repay him with the results.
"In the last couple of games they haven't been what this football club should have."
Arsenal host Aston Villa in the Premier League on Saturday.