It looked as if the points might be heading north when they visitors - making their first ever competitive visit to Broadwood - snatched the lead late on.
But Danny Lennon's side hit back quickly through David Goodwillie to earn a share of the spoils.
Clyde looking to follow up their Scottish Cup win at Ayr, started brightly and Lewis Jamieson was just wide with a chipped effort after being played in by Michael McGlinchey.
Cove keeper Stuart McKenzie kept out a Ross Cunningham shot while the visitors' best first-half chance fell to Rory McAllister who was denied by David Mitchell.
Clyde had loud claims for a penalty turned down after a Jamieson shot appeared to strike a Cove defender's arm.
The Bully Wee then fell behind in the 76th minute when Connor Scully curled the ball beyond Mitchell from the edge of the box.
But just two minutes later Clyde were back on level terms, Saturday's goal hero Barry Cuddihy seeing a shot blocked but playing the rebound into the path of Goodwillie who made no mistake from close range.
Both clubs had chances to grab a late winner but there was no further scoring.
The result kept Clyde out of the relegation play-off place; they remain a point above second bottom Dumbarton.
Next up, on Thursday, is a home match with bottom side Forfar, followed by Saturday's trip to play leaders Falkirk.