Billy Davies' side triumphed 2-1 courtesy of the man on loan from Polonia Warsaw, who struck in the 103rd minute.
Luke Chambers became the first man to beat Danny Coyne this season as he netted a second-half equaliser.
It was Middlesbrough who had gone into the interval with a single goal advantage, following a precise strike from Adam Johnson in the 43rd minute as he latched onto Marvin Emnes' perfect pass down the right before lashing the ball low into the net from the corner of the box.
But had the home side demonstrated the same clinical nature in their finishing, they could easily have been ahead themselves by that stage.
An early shot from Majewski was deflected, looping narrowly wide and Dexter Blackstock's control had let him down at the vital moment as he threatened to break clear following a rampaging run from defender Wes Morgan.
The normally lethal Robert Earnshaw then twice fired half-chances wide of the post when he might have expected to test the keeper.
Paul Smith, enjoying a rare start in the Forest goal, had to make smart saves to deny Johnson and Jeremie Aliadiere as Middlesbrough posed a considerable threat on the break, while Mark Yeates fired wide of the post.
But it was a familiar tale for Forest, who had already paid the price for a lack of cutting edge during a start to the season that is yet to bring a Championship victory.
Fortunately, following a lively start to the second half, a demonstration of how to finish from an unlikely source was to help Forest back into the game, as Chambers produced an improvised back heel to guide an Arron Davies corner high into the net on the hour mark.
After drawing level, it was Forest who looked the more dangerous of the two sides, forcing a succession of corners and being denied a penalty claim after Blackstock had gone down under pressure from Taylor.
The dangerous Johnson had also come agonisingly close for Middlesbrough as the game edged into extra-time.
But it was Majewski who popped up to secure Forest's progression into the next round after firing confidently beyond Coyne after substitute Dele Adebola had laid on the chance.