Coyle has overseen a disastrous start to the season.
But Carlisle, who played under Coyle at Burnley, told talkSPORT: "It would be insane to sack Owen now.
"I have been saying this to all and sundry - they are a victim of fixtures so far this season. In their first seven games they've played many teams that finished in last season's top six and that's a hard task for anyone.
"I have no doubt that they will start to pick up points and they will play competent, attractive football. They will be expected to win a few of their matches coming up now.
"Perhaps, if they are left with three points after 15 games then maybe something will need to be said but suffice to say, they should stick with Owen."