The defender thundered in a shot from the edge of the box in stoppage time to grab his first goal of the season.
The Binos had a goal disallowed in the first half with a Daniel Jardine header.
But the former St Johnstone youth was not to be denied when he gave Stirling the lead in the second half.
It looked like the Binos were going to hold on until Cogill fired in late on.
Clyde remain in third in League Two while Albion are in eighth.
Both teams would have been hoping to build on last week’s victories as Danny Lennon kept faith in the side which ended league leaders Edinburgh City’s 12-game winning streak last week, while Darren Barr was the only change for Stirling Albion under new boss Kevin Rutkiewicz.
There was not much in the way of opportunities for either side early on, but Stirling did have the ball in the back of net through a Daniel Jardine header. However, the celebrations were cut short as the linesman flagged for an infringement in the box.
At the other end, John Rankin had a speculative effort from 25 yards with a dipping half volley which was comfortable for Binnie, before Dominic Docherty volleyed straight at Blair Currie for Stirling.
The pendulum continued to swing from end to end with Clyde defender Cogill getting the faintest of touches on a inswinging Rankin cross, which came back off the post. However, McStay could only fire the rebound into the side netting.
Pressure was building for Clyde, Syvertsen showed a great piece of play, skipping in between McGeachie and Horne before cutting the ball back for McStay, but his effort was blocked by Marr.
Clyde started the second half brightly and Rankin brought another good save from Binnie from a 25-yard free kick.
However, it was the Binos who broke the deadlock on 56 minutes. Mackin’s headed effort from McIndoe’s cross was saved by Currie, but Jardine was there to tap in the rebound for his second goal in as many games.
Danny Lennon threw on Ross Lyon and Karim Belmokhtar in search of an equaliser and Lyon looked to make an immediate impact but the sting was taken out of his shot from the edge of the box.
The Bully Wee continued to push for an equaliser and they levelled into stoppage time when Cogill rifled home from the edge of the box through a ruck of players, after Stirling failed to clear their lines.