A second-minute strike from Graeme Holmes kept 10-man Alloa in second place in the Scottish Third Division.
Holmes was on hand to convert the rebound after a shot from Kevin Cawley had come back off a post.
The visitors were reduced to 10 after only 32 minutes, however, when Fraser Forrest was given a second yellow card for a second deliberate handball.
Clyde keeper Filip Mentel made two fine saves and Jack Sloss hit a late chance to equalise over the bar from 10 yards.
Clyde: Mentel, Brown (Fitzpatrick 75), McQueen, Declan Gallagher, Sharp, Neill, Sweeney, Gray (Oliver 45), Ryan Kane (Sloss 46), Cusack, McDonald
Subs Not Used: McCue, Scullion
Alloa: McDowall, McCullagh, Forrest, Harding, James Doyle, Masterton (Ross McCord 42), Young, Ryan McCord, Cawley (Wright 71), Winters (Docherty 33), Holmes
Subs Not Used: Bain, One
Sent Off: Forrest (32)
Booked: Forrest, Ryan McCord
Goal: Holmes 2
Ref: Euan Anderson