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 impressive Young single-handedly keep the Shots alive before the break. First, Marvin Emnes picked out Stuart Parnaby for a treacherous effort after reaching Richard Smallwood's through-ball at the by-line, but Young stood firm.
The Brisbane-born goalkeeper then inspired applause from all sides of the Riverside four minutes before half-time, springing off his line to palm Merouane Zemmama's curling strike away from the top corner.
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 an Adam 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.
- Sports & Recreation
- Danny Hylton
- Jamie Young
- Dean Holdsworth