The visitors were on course for a win which would have lifted them up to fourth in table following Lawrence Shankland's fourth-minute opener.
The defender, a summer recruit from Livingston, is becoming a goal threat for Accies as that was his third of the season.
Hamilton head coach Brian Rice made four changes to the side which lost the previous weekend at Kilmarnock with Lee Hodson, back after a period of self-isolation, Kyle Munro, Andy Winter and Tunde Owolabi coming into the starting line-up.
Dundee United manager Micky Mellon stuck with the same team which defeated St Mirren and, just as in that game, Shankland opened the scoring.
The visitors took just four minutes to break the deadlock. While Shankland's goal against St Mirren was a spectacular effort, this was a far more routine finish after some fine build-up play.
Logan Chalmers burst down the right flank and his cross was collected by Adrian Sporle. His route to goal was blocked but his cut-back was perfect for Shankland to finish off from close range.
Chalmers should have doubled Dundee United's lead in the 16th minute when he was played in by Clark but he blazed high over the crossbar.
Owolabi spurned a glorious chance to draw Hamilton level on his first start for the club since joining from FC United of Manchester. Munro sent in a wonderful cross but the unmarked striker glanced his header wide.
Luke Bolton then went close for Dundee United when the full-back cut inside onto his left foot and his angled drive went just the wrong side of the post.
Five minutes before the break Owolabi had another opportunity as he got on the end of a David Templeton free-kick but again he sent his header wide.
Hamilton made two changes at half-time with Ross Callaghan and Marios Ogkmpoe coming on for Reegan Mimnaugh and Winter. And nine minutes into the second half Dundee United made a switch with Mark Connolly replacing Sporle.
Hamilton goalkeeper Ryan Fulton made a vital save on the hour mark to deny Clark. Shankland played in his strike partner but Fulton darted off his line and pulled off a crucial stop.
With 15 minutes remaining Hamilton drew level through Odoffin.
Templeton swung in a free-kick, after Ryan Edwards had fouled Ogkmpoe, to the far post where the defender stooped down to guide a diving header beyond Benjamin Siegrist and notch Accies' first goal at their own patch this season and his third of the campaign.
Scott Martin went close with a strike from 20 yards for Accies and then, in the closing stages, Clark fired over the crossbar from close range.