Clyde's win over relegated Stirling Albion was not enough to escape a play-off to avoid following their opponents into Division Two next term.
Gary McSwegan pounced just before half time when goalkeeper Scott Christie spilled Dave McKay's long ball.
Pat Clarke, unmarked at the back post, converted Ruari MacLennan's pin-point cross just after the break.
Clarke's late penalty gave Clyde their best win of the season, but Morton stayed ahead on goal difference.
It means the Cumbernauld outfit must face Alloa Athletic, who finished fourth in Division Two, in the first leg of the semi-final play-off on Wednesday.
Clyde: Hutton, Masterton, Higgins, Bestvina, William Gibson, McKay, James Gibson (Wilson 36), Ruari MacLennan, Albertz (Michael McGowan 46), McSwegan (Arbuckle 48), Clarke. Subs Not Used: Bradley, Cherrie.
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