Josh Morris was the man with the honours for the home side, lashing home the first goal after Derick Williams had blocked Dobbie’s initial effort.
But Fleetwood could not keep hold of the their advantage before the break.
Kieran Agard had already seen one shot superbly saved by Chris Maxwell and another fly past the post when just after the half-hour mark he found himself on the end of Joe Bryan’s cross and stabbed the ball home from close range.
Fleetwood were still napping when Agard bagged his second two minutes after the interval.
A neat backheel by Wade Elliott set up Luke Freeman to cross, Aaron Wilbraham’s first shot was saved by Maxwell and Agard pounced on the loose ball to drive the ball home.
City struck again five minutes later with Mark Little’s cross flicked over Nathan Pond by Agard, leaving Wilbraham to apply the finish.
Fleetwood brought on Jamie Proctor in the hope of battling back and the substitute made an impact within a minute, heading home Morris’ cross to cut the arrears.
Conor McLaughlin could have provided a late leveller, Frank Fielding’s fingertips denying the Northern Ireland international his first-ever Football League goal.
But two minutes from time the home side did find a way back.
Matty Hughes’ effort was blocked and the ball fell to Dobbie, who let fly with an unstoppable curling effort to complete the fightback.
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