Clyde were forced to fight all the way against a side two divisions below them before disposing of Montrose and moving into the Scottish Cup fourth round.
Yet the First Division side had taken the lead after eight minutes, when Gary Arbuckle forced the ball home.
Gary Wood and Steven Doris passed up chances as Montrose dominated after the break, while Clyde goalkeeper David Hutton denied Andy Rodgers.
But Dougie Imrie scored in the dying seconds to ensure Clyde's passage.
Clyde: Hutton, McGregor, Smith, Higgins, McKeown, MacLennan (Carrio 58), Cardle (Valentim 81), MacLennan, Arbuckle, Imrie, Michael McGowan.
Subs Not Used: Cherrie, Kirkup, David McGowan.
Goals: Arbuckle 8, Imrie 90.
Montrose: Peat, Cumming, Gardiner, McLeod, Dobbins, Davidson, Doris (Gates 78), Gibson (Maitland 83), Wood, Black (Rogers 61), Stein.
Subs Not Used: McKenzie, Thomson.
Booked: Dobbins, McLeod.
Att: 821
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