Blues' victory could have been by a bigger margin, with some great chances going begging and an effort ruled out following a foul in the Sky Bet Championship contest.
Birmingham boss Lee Bowyer handed debuts to Tahith Chong, Ryan Woods and Matija Sarkic, while fellow new signings Chuks Aneke and Juan Castillo were included among the substitutes.
Chong was soon involved, firing in a shot which was blocked by John Egan inside the opening minute.
David McGoldrick threatened to find a way through at the other end before seeing his shot charged down just inside the area.
Colin made the decisive breakthrough in the 19th minute, rising in the area to meet a cross from Jeremie Bela which evaded Ben Osborn.
United went close to making an almost immediate response when Sander Berge slipped the ball to Oliver Burke on the right-hand side of the area and his low, angled shot struck the near post.
As the hosts looked to get back on level terms before the break, Lys Mousset sent over a low cross from the right which found John Fleck but the midfielder's side-footed effort was deflected wide.
Oliver Norwood then saw an opportunity to try his luck from distance and his shot went narrowly wide of Sarkic's right-hand post.
Early in the second half, Burke broke free on the right and tried to find Mousset, who was running in, but the ball was too far in front of the striker, allowing Sarkic to cut out the danger.
Mousset fell awkwardly while stretching to reach the ball and had to be helped from the pitch after receiving treatment, with Rhian Brewster coming on in his place.
Lukas Jutkiewicz wasted a great chance to put Birmingham further ahead, heading wide of the goal.
Scott Hogan then spurned an even better opening, lifting the ball over the bar from inside the six-yard box after getting on the end of a Jutkiewicz knock-down.
In between those two efforts for the visitors, McGoldrick curled a shot just wide.
Colin thought he had scored his second goal of the match shortly before the hour but his header was ruled out after a push by Jutkiewicz on Osborn.
United players appealed for a penalty when Fleck's drive struck the arm of an opposition defender, sending the ball wide, but they were only awarded a corner.
Jokanovic made a second attacking change in the 69th minute, sending on Billy Sharp in place of Fleck.
A third change up front 10 miinutes from time saw Oli McBurnie replace Burke.
In the opposition line-up, the lively Chong was replaced by Ivan Sunjic and Castillo took over from fellow debutant Woods.
With Birmingham defending in numbers, United never seriously threatened to level in the closing stages, going down to a defeat on their return to the Championship.