Kevin Watt's close-range strike put the visitors ahead after 30 minutes after Montrose struggled to clear a corner in the swirling wind.
Three minutes later, Clyde made it two when Keiran MacDonald crossed for Scott Ferguson to head home.
Ross McCord's powerful shot produced a fine save from Clyde goalkeeper Nic Rajovic as Montrose tried to come back, but hopes of salvation were dashed when McCord was shown a second yellow card for dissent a minute before half-time.
The Cumbernauld club prospered from Peterhead slipping to a 1-1 draw with Albion Rovers at Balmoor, in a match that only went ahead after a late pitch inspection.
Peterhead's Graeme Smith made two decent saves from Rovers' Liam Cusack in the first half.
With seven minutes left, Mark McGuigan's low strike gave Rovers the breakthrough, but the Coatbridge side could not hold on and Ryan McCann's 86th-minute header rescued a point for Peterhead.
Berwick Rangers emerged 2-0 winners against East Stirling in an ill-tempered encounter.
Darren Lavery scored his ninth league goal this season to give Berwick a 10th-minute lead and Lee Currie drove home a late second to seal the points after the visitors had had Michael Bolochoweckyj and David Greenhill sent off.
Two fixtures fell victim to the weather: Annan Athletic v Queens Park and Stirling Albion v Elgin City.