The home side opened the match brightly and Gary Arbuckle was on target in the first minute but his shot lacked power and was easily saved by Terrors keeper Euan McLean.
Shortly after Steven Masterton released Michael McGowan but his low cross into the box was cut out by Darren Dods.
United's response was a dangerous cross from Noel Hunt towards the head of skipper Robson but keeper David Hutton made a good save to deny the Terrors man.
Further danger for the visitors came from an inswinging corner taken by Dougie Imrie which was blocked at the near post by United's Morgaro Gomis.
Hunt then went close for the visitors when he headed a Sean Dillon cross just wide.
Danny Grainger hurled a long throw into Clyde's six-yard box for Jordan Robertson to gain control but he was crowded out before he could find space for a shot.
Clyde were then awarded a free-kick on the edge of the Dundee United penalty area but Masterton's powerful drive was blocked by the well organised wall.
Hunt was lucky to escape a yellow card for going down too easily looking for a penalty.
United had a great chance to break the deadlock three minutes before the break but Willo Flood headed Robson's cross wide when it looked easier to score.
Two minutes after the restart United's Lee Wilkie misjudged a long ball out of defence leaving Arbuckle with a clear run on goal.
However, he elected to shoot from a tight angle when team-mate Kevin Bradley was better placed to take a pass in front of a gaping goal.
Clyde then conceded a free-kick in a dangerous area and were thankful for keeper Hutton who turned Flood's fiercely-hit shot for a fruitless corner.
The goal that secured United a fifth round trip to face St Mirren came after 56 minutes.
There seemed little danger to Clyde when Robson picked up a loose ball 30 yards out but, when the home defenders backed off, the United captain struck a 30-yarder into the top corner leaving Hutton helpless.
There were half-chances at either end as the game wore on but United never looked like losing their slender advantage.