The striker, drafted in until the end of the season from Doncaster in midweek, came on as a half-time substitute and turned the ball home after Alex goalkeeper John Ruddy had spilled his first effort.
Lions top-scorer Neil Harris was involved in most of the first-half's high points before going off with what appeared to be concussion.
After seven minutes Ruddy and defender Julien Baudet both left a long ball to each other and Harris nipped in only to see his toe-poke goalwards drift narrowly wide.
He should have scored in the 37th minute after being left unmarked when James Henry's corner was delivered but he flicked his close-range header wide.
Price flashed a header just wide of the upright as time ran out but was on alert enough to pounce on the loose ball in added time to win it.