McCulloch scored his 25th goal of the season to break the deadlock 10 minutes into the second half before Hutton doubled the lead a minute from time.
The away side formed a guard of honour before kick-off as they paid tribute to a Rangers side playing their first game at Ibrox since securing the Division Three title.
Ally McCoist's side dominated from the outset, with McCulloch just failing to connect with a ball across goal from Lee Wallace.
Clyde goalkeeper Jamie Barclay was then forced to save a deflected Hutton shot that came off McCulloch and was called into action again to keep out a swerving drive from Fraser Aird.
But the home side stuck to their task and took the lead when McCulloch got on the end of a ball from Templeton to score the opener.
Young defender Luca Gasparotto went close to adding another only to be foiled by the busy Barclay.
Rangers made sure of the three points late on though courtesy of Hutton after the midfielder fired an angled shot past Barclay and into the far bottom corner.