Gary McSheffrey’s side were good value for their victory, producing some neat, tidy and attractive football in stages.
They established an early lead thanks to Goodman’s well-executed strike finish from inside the box.
Lincoln were forced to play the majority of the game with 10 men after their goalkeeper was sent off for deliberately bringing down Owen Scattergood when the striker was put clean through on the quarter hour mark.
Goodman soon made it 2-0 with a ferocious 25-yard shot after the Imps were caught cold in their efforts to play out from the back, while under 16s midfielder Will Flint and right wing back Will Hollings both went close with further attempts.
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The Imps didn’t give up though and, having already struck the woodwork before half time, were thwarted again when Bobby Faulkner produced a last-ditch goal-saving challenge shortly after the re-start.
Any hopes that Lincoln harboured of a comeback were dashed on the hour mark, however, when Cole turned the ball into an empty net to make it 3-0 after Hollings had intercepted a stray pass high up the pitch.
Scattergood then saw a header saved and Michael Nesbitt planted a low free-kick narrowly wide having been hacked down following a marauding run to the edge of the box as Rovers looked to add more goals.
The hosts struck twice from set pieces in the dying minutes to add some respectability to the scoreline.
Rovers’ youngsters continue their season with a home clash against Grimsby Town on Saturday.