Boro were held at arm's length by Dean Holdsworth's Shots for 83 minutes at the Riverside Stadium, but Jutkiewicz made the breakthrough with a forceful header.
But Danny Hylton extended his run as the competition's top scorer with a dramatic goal on the counter-attack to tee up a replay at the Recreation Ground, only for Jutkiewicz to break the spirited visitors' resolve with a goal at the death.
The first half was largely a battle between invention and physicality as Aldershot, marshalled by goalkeeper Jamie Young, stubbornly repelled the hosts' fluid raids.
The hosts sought to take the initiative and Adam Reach's mazy dribble fell two moves short before Aldershot broke quickly.
Hylton tumbled on the edge of the box under the attentions of Stuart Parnaby and Danny Rose forced Jason Steele into a fine fingertip save from the resulting free-kick.
But Reach would make his skill on the ball count moments later, dancing past the Shots defence to pick out Marvin Emnes, whose lay-off to Merouane Zemmama sent Young sprawling.
Hylton remained unpredictable, though, and Seb Hines' decision to clatter the onrushing striker on the edge of the box earned him the first yellow card of the game.
Peter Vincenti's free-kick left much to be desired and soon Young was snaffling a bullet header from Jutkiewicz at the other end. The impressive Young would single-handedly keep the Shots alive before the break.
But Boro were a constant menace in the Shots box and it fell to Jutkiewicz to make a belated breakthrough seven minutes from time. The ex-Coventry striker rose highest to head home a Reach cross which was already drifting towards the top corner.
Boro's joy was short-lived, though, as Hylton skipped on to an incisive through-ball to slot home his eighth cup goal, setting up a frantic finale.
The final test for Holdsworth's plucky League Two outfit would be six minutes of stoppage time, but they could not hold out, Jutkiewicz comfortably sliding the ball beyond Young to settle the tie.