Substitute Ian Poveda, on loan from Leeds, tapped home for Blackpool’s second win in three in the 74th minute after a dreadful mix-up between defender Kevin Long and goalkeeper Neil Etheridge.
“We’re still alive, and in that dressing room we’re starting to believe we can do it,” said Dobbie.
“I told the players we’d got nothing to lose and that we needed to go for it so could we get the goal to get us the win to keep fighting until Friday night before the other teams play.”
Birmingham dominated much of the first half then had four shots cleared off the line in the space of a few seconds before the hour.
But they were unable to break the deadlock despite numerous chances.
A routine shot from Tahith Chong took a deflection but it was spilled by goalkeeper Daniel Grimshaw, who gathered at the second attempt.
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