A Paul Dixon own-goal four minutes from time ended Dundee's unbeaten home record and lifted Clyde off the foot of the table.
Clyde came closest to scoring in the opening half when Dougie Imrie's corner was cleared off the line.
Dundee raised their game after the break but never really tested goalkeeper David Hutton.
And they were undone in the 86th minute when Imrie's low driven cross was turned into his own net by Dixon.
Dundee: Samson, Worrell, Dixon, McHale, MacKenzie, Palenik, Swankie, Robertson (O'Brien 83), Lyle, Sturm (Daquin 72), Malone. Subs Not Used: Griffin, Rae, Roy.
Clyde: Hutton, McGregor, Wilson (Smith 37), Higgins, McKeown, Masterton, David McGowan (Arbuckle 72), MacLennan, Cardle (Bradley 79), Imrie, Michael McGowan. Subs Not Used: Bestvina, Cherrie.
Booked: Michael McGowan.
Goals: Dixon 86 og.
Ref: K Clark (Scotland).