A first-half double from skipper David Goodwillie eventually proved enough to secure the points – although Danny Lennon’s side did have to survive a hectic finale which saw them concede a penalty, lose a goal and have a player sent off.
The Bully Wee didn’t have their troubles to seek pre-match as Joshua Bradley-Hurst was drafted in at short notice to make his league debut in goals with both regular number one David Mitchell and new signing Neil Parry unavailable.
Lennon also made changes in defence, going with a back three as Adam Livingstone and Conrad Balatoni replaced Billy Mortimer and Barry Cuddihy.
Perhaps understandably Cove started the stronger, forcing a series of early corners with Ross Cunningham having to head a Jevan Anderson header off the line.
But it was the Bully Wee who struck first with eight minutes on the clock.
Balatoni's through pass split the visitors' defence and found Goodwillie racing clear to collly steer his finish beyond former Clyde keeper Kyle Gourlay.
Goodwillie headed over when a Morgaro Gomis cross found him unmarked and Cunningham then ran at the Cove defence before firing in a low drive which eluded Gourlay but smacked off a post.
Bradley-Hurst was called into action, safely gathering a Mitch Megginson drive and then doing well to turn away an Anderson header.
Clyde then doubled their advantage just before the interval, Goodwillie robbing a too casual Iain Vigurs and bearing down on goal before lashing the ball home after Gourlay had stopped his initial effort.
Cove pushed forward in the second half to try to get back into the game but it wasn’t until six minutes from time that they reduced the deficit.
That set up a frantic finale and Megginson had a great chance to level but Bradley-Hurst denied him with a terrific stop.
D eep into stoppage time Matty Elsdon was red carded for a last man tug on McAllister.
But Jamie Masson fired the free-kick over the bar and Clyde held on.