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But the home side were soon level, with Simon Yeo chipping Matt Glennon when going one on one.
Minutes later Yeo set up David Freeman to put the Imps in the lead, and by half time it was 3-1 with Yeo on target again.
Just before the hour mark Allan Pearce added to their lead, latching onto a cross-field pass then firing past Glennon.
Tigers midfielder Scott Kerr was then forced to withdraw after an Adam Buckley challenge left him injured.
Clayton Donaldson, on for the injured Kerr, got a consolation for City with 20 minutes to go, capitalising on Paul Crager's slip to pull it back to 4-2, but Lincoln were too good for City, who now lose their undefeated record in the Avon Insurance League this season.
City line-up: Glennon; Heard, Chapman, Wicks, Holt; Williams (Russell 79), Kerr (Donaldson 66), Philpott, Peat; Alexander (Turnbull 79), Bradshaw.
Un-used subs: Harvey, Wiseman.