A brace each from Dylan Nguene Bikey and Peter MacDonald means these League 2 rivals will need a replay to determine who progresses to the fifth round.
But MacDonald and Clyde will feel they should already be through after the striker’s second-half penalty was saved by Cammy Binnie.
Chances were shared during a lively opening with Connor McLaren heading narrowly wide for Albion before Clyde responded through a David Gormley shot which was well saved by Binnie.
Stirling made the breakthrough when Frenchman Nguene Bikey showed his eye for goal when he took a pass from McLaren, wriggled free and fired home with the aid of a deflection.
The Bully Wee fought back and deservedly equalised when MacDonald latched on to Gormley’s pass and beat Binnie with a low shot from 10 yards.
Stirling almost restored their lead just before the break when Willie Robertson’s powerful shot was deflected inches wide.
Binnie was in action to deny MacDonald his second with a smart save moments after the restart but chances became a rarity as a series of niggling fouls punctuated the opening exchanges of the second period.
However, the experienced MacDonald was eventually rewarded with a second goal, the striker firing an angled shot in at the left-hand post.
MacDonald was handed the perfect chance to clinch it for Clyde when he was brought down in the area by Binnie but the keeper atoned by saving the striker’s spot-kick.
Instead honours were shared when Nguene Bikey pounced with the equaliser six minutes from time.