The Minstermen dominated the opening half but failed to take full advantage and led only by a single-goal at the break.
That arrived ten minutes prior to the interval when Brodie collected Alex Lawless' through ball before being taken down while attempting to round goalkeeper Chris MacKenzie, who then guessed correctly but couldn't keep out Brodie's penalty.
And they were shocked early in the second-half when Matt West glanced a header over Michael Ingham to equalise before the prolific Brodie restored home advantage just beyond the hour.
Hinckley were then given another route back into the contest with seventeen minutes remaining after substitute Ben Purkiss was penalised for a foul inside the area. But West struck his penalty high and against the crossbar.
Appropriately it was Brodie who eventually settled the tie in the final minute when completing his hat-trick.
Recalled midfielder Neil Barrett had already tried an audacious lob inside the first minute, while both James Meredith and Brodie (twice) also threatened. Brodie then went down in the area under the challenge of MacKenzie, but on this occasion no reward was forthcoming.
Hinckley striker West soon drilled wide from 30-yards in reply, but the visitors were again under pressure on the half-hour as Brodie collected Rankine's pass and swept past the keeper before his attempted finish was blocked on the line.
However, Brodie opened the scoring from the penalty spot moments later, while Levi Mackin hammered a 25-yard effort against the crossbar six minutes before the break. Hinckley responded by forcing home keeper Ingham to turn a free-kick over before then falling low to save from Alex Taylor.
The Minstermen still ended the first-half well in the ascendancy with Barrett somehow heading wide at the far post following superb work from Lawless. Rankine then had a 20-yard drive fended behind by MacKenzie, who twice blocked at close-range from the subsequent corner.
But the visitors claimed a shock equaliser three minutes into the second-half when Taylor's cross from the left appeared to glance off the head of West at the near post and beyond the helpless Ingham.
That seemed to clearly lift Hinckley and they proceeded to enjoy a substantial amount of advanced possession, during which a couple of efforts were sent wide of the target, while Ingham was required to gather a dangerous ball into the area.
York then reclaimed their lead just past the hour when Barrett supplied the pass for Brodie to expertly draw the keeper. Rankine almost added a third goal moments later when his shot produced a fingertip save from MacKenzie.
Hinckley were though still very much in the tie and on loan striker Sam Smith sent an effort over the target before they were awarded a penalty on 73 minutes after Purkiss was adjudged to have fouled just inside the area. However, West's rising spot-kick was thumped against the crossbar.
To their credit the visitors kept going until the end and Ingham was twice called upon to block, while his opposite number MacKenzie denied Rankine at his near post.
And it was Brodie who suitably resolved the issue in the final minute when switching possesion onto his right-foot and curling his finish past the valiant MacKenzie and in off the far upright.
York City: Ingham, Parslow (Purkiss 60), McGurk, Graham, Meredith, Lawless, Mackin, Barrett, Carruthers, Rankine (Gash 85), Brodie
subs (unused): Mimms, Ferrell, Gall
Goals: Brodie (pen 35, 61, 90)
Booked: Meredith
Hinckley United: MacKenzie, Cartwight (Mason 62), Collins, Lister, Franklin, Hall, Dillon, Taylor (Newton 80), Roberts, Smith, West
subs (unused): Hamilton, Young, Lancaster
Goals: West (48)
Booked: Hall, Lister, Dillon
Ref: R. Fletcher (Melbourne)
Att: 853