David Ajiboye opened the scoring for the hosts in the 62nd minute and they looked on course for all three points until Carl Piergianni drew the Latics even in the 88th minute and Junior Luamba won it for the visitors in the sixth minute of stoppage time.
Story of the match
Sutton began the match on the front foot and they had a chance in the fourth minute as Robert Milsom sent in a cross that came to Lewis John, but his header went over.
The hosts had further chances, but a header by Omar Bugiel was too high and Enzo Boldewijn shot from just outside of the Oldham penalty box, missing to the left.
Bahamboula would have an even better chance seven minutes from halftime as he forced a good save from Yellow Army goalkeeper Dean Bouzanis.
Boldewijn and Ajiboye saw their attempts in first-half stoppage time blocked as the hosts continued to pile on the pressure. Craig Eastmond then saw his shot stopped by Latics netminder Dean Leutwiler.
In the 56th minute, Luamba tested Bouzanis, but he was able to save. At the other end, Eastmond hit the bar with a shot and Ajiboye had his chance blocked.
The midfielder would find Sutton's breakthrough six minutes later as he finished past Leutwiler from close range for the Yellow Army's first-ever goal as a Football League club at their home ground.
Eastmond and Donovan Wilson were looking to add a second, but saw their shots blocked. In the 88th minute, Oldham leveled as Carl Piergianni beat Bouzanis in the bottom corner on a cross from Davis Keillor-Dunn.
The match appeared to be headed for a draw until Keillor-Dunn turned provider for the Latics' winner, crossing for Luamba, who beat Bouzanis with the last kick of the ball and broke Sutton's heart.
Man of the match: Junior Luamba
He had a couple of chances before his match-winner as Oldham grew into the game in the second half. The goal that gave the Latics all three points and the dramatic fashion in which he scored it may be looked back on if they turn their season around.