The Polish midfielder struck with a little over 10 minutes left, latching on to Andy Reid's pass to fire past City goalkeeper Joe Anyon.
Forest, with summer signings Reid and George Boateng in a strong-looking starting XI, also handed a debut to trialist Anthony Gardner, but had to wait until a raft of City changes before they finally forged ahead.
An early 25-yarder from Reid tested Anyon's palms early on, but Lincoln nearly went ahead after 10 minutes when Jamie Taylor latched on to a goal-kick, but snatched his shot wide of Lee Camp's right-hand post.
Adam Watts came to Lincoln's rescue midway through the first half, clearing a Reid shot off the line as both sides threatened to cancel each other out.
After the break, Boateng brought a save out of Anyon, low down to his right, before City's Simon Russell cracked a shot narrowly over the crossbar.
Forest boss Steve McClaren cut a frustrated figure on the sidelines as the Reds failed to find a way through a resilient Lincoln defence, but after several substitutions by the home side that upset their rhythm, Majewski arrived on to Reid's pullback to smash the ball past Anyon from 10 yards and win it for the Reds.