All the goals came in a fabulous first half, but this was compelling to the end and either side might have snatched it in the closing stages.
Stirling, unbeaten in the league since the opening day, started strongly and Ross Forsyth had an effort cleared off the line before they took the lead after eight minutes.
Innes Ritchie man-handled Colin Cramb in the box and Chris Aitken made a splendid job of the resulting penalty.
The visitors looked in control but Cowden turned the game on its head with two goals in as many minutes.
First Markus Paatelainen headed Martyn Fotheringham's corner down and over Myles Hogarth then Pat Clarke was given too much space on the left and he teed up Fotheringham to score with a crisp first-time shot.
Albion hauled themselves level in the 25th minute when Marek Tomana released Cramb and he thumped a superb shot into the corner.
Stirling continued to look strong and after 38 minutes Aitken fired just over after being set up by Cramb.
Understandably the second half was a more tame affair with few chances, although Clarke missed a great opportunity after 63 minutes lobbing the ball over when clean through.
The same player came close again with a similar attempt after 79 minutes, while David Orr pushed out a fierce Aitken drive at the other end.