New signing Kevin Hemagou struck from close range to keep Lincoln's outside chances of making a play-off spot alive after Carlton failed to deal with a deep ball into the area in the seconnd minute of added time.
The Millers could have no complaints after a below-par performance but will be kicking themselves for not picking up at least a point.
Matt Cotton gave the visitors a second minute lead when his free-kick was misjudged by home `keeper Jack Steggles and flew into the net. Carlton levelled in the 58th minute with their best move of the game, Tyler Blake scoring his first goal for the club on his full debut after a cross by Jordan Wilson.
Image courtesy of nonleague.pitchero.com
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