Coleman believes it is time for them to step up and ensure the former Hull and Watford boss is a success with his new side.
"David Meyler has had him (at Hull) and said he's a great manager," he said. "So it's up to us now at the club to kick on and try to do really well because we're chopping and changing managers quite a lot, so it's up to the players now to step up and make sure this manager stays in the job.
“It's always very easy for us players to hide behind managers. We have had three or four now and maybe we get away with a bit of the stick that's flying around because it's always on the manager's toes, whether that be Roberto, Ronald, Sam, Moyesy before that.
“It's about time us players now step up to the plate. We've not been close to winning something for a long time and at Everton that's what we want, that's what I want.
“I've been there a long time and, if I leave without having won anything, I'll be very disappointed. As a group of players, no matter who the manager is, we need to take responsibility."