Colin Calderwood's side had only a Kris Commons strike to show for a first half in which they had dominated, but striker Tyson found the back of the net twice in the first 15 minutes of the second half to put the match beyond Peter Jackson's Imps.
Tyson had seen Alan Marriott make an outstanding save to deny him, as he palmed a low shot around the foot of the post in the sixth minute and saw another effort ruled out for offside, after he had found the bottom corner.
But the Forest striker was left with his head in his hands after firing over a virtually open goal from six yards, after being picked out by a superb cut-back from Julian Bennett.
Forest continued to carve out opportunities, in the same manner as they had done against Crewe, with Commons trying his luck from the edge of the box, only to see two shots in quick succession blocked by desperate defending.
The midfielder then came even closer as, after Tyson had burst powerfully down the left, he looped a header narrowly over the bar.
Lincoln came close to snatching the lead themselves, as Louis Dodds charged into the box, only to be denied by a superb tackle from Bennett, who thwarted a scoring chance but injured himself in the process and was replaced by James Perch.
And the enforced change failed to put Forest out of their stride as, within three minutes, they were finally in the lead.
Commons burst forward down the left, exchanged passes with Chris Cohen and cracked a left-foot shot, from the edge of the box, into the far bottom corner.
Forest went in search of a second and saw Marriott deny Arron Davies, as he got his body in the way of a powerful, driven effort, while Commons dipped a shot just over.
Lee Frecklington, Lincoln's most dangerous player, forced a capable save from Paul Smith, but it was Forest who were on top.
And, after the interval, they were to cap off their dominance with two quick goals - both from Tyson - to kill off the game.
His first was a solo effort rolled calmly inside the post, beyond the advancing keeper, after his pace had carried him into the box from the left.
While the second was a team effort, with Lewis McGugan and Davies combining down the right, with the Welshman then sending a teasing ball across goal for Tyson to slide home from close range.
Lincoln pulled one back through the persistence of Jamie Forrester, who danced into the box and deserved his good fortune as his shot deflected through a melee of players and beyond the wrong-footed Smith.
But it was Forest who progressed through to the second round, where a trip to Luton Town awaits them.