For 84 minutes, the Ayrshire juniors were clinging to hopes of taking Hearts, fourth in the SPL, to a replay, until Gordon Smith netted following a defensive mix-up.
Hearts debutant Fraser Mullen had a 20th-minute penalty well saved by Andrew Leishman, who also denied Rudi Skacel and substitute David Templeton as the hosts' casual approach nearly cost them.
Hearts defender Darren Barr had to be taken off on a stretcher with his neck in a brace after a heavy landing midway through the second half.
In the fifth of eight minutes of stoppage time, Gordon Pope thought he had netted an equaliser only for the strike to be ruled out for offside.
Manager Paulo Sergio has predicted a difficult month for Hearts, with player sales inevitable in order to ease the financial burden on the club and to satisfy the Scottish Premier League's demands over the payment of player wages, which were delayed in each of the last three months.
The outlook on the field is concerning if Hearts are going to be forced to rely on youth.
After three wins in four SPL games, Sergio made eight changes for the visit of Auchinleck, who were roared on by a sizeable support which almost filled the away end.
If those Hearts fans in attendance were expecting the most one-sided match at Tynecastle since Tottenham won 5-0 in August, but this time in the hosts' favour, they were to be disappointed.
Auchinleck were camped inside their own half but proved difficult to break down, with Hearts' best route appearing to be out wide through Arvydas Novikovas and Andrew Driver.
Driver was harshly booked after 10 minutes for simulation but Hearts had a penalty soon after when Novikovas bamboozled Pope and was brought down.
But 18-year-old right-back Mullen's weak shot was turned around the post by Leishman as Hearts missed from the spot for the fourth time in succession.
Ian Black shot well over and Scott Robinson was booked for a wild lunge on Keir Milliken as Hearts' frustrations began to tell.
Novikovas, whose tendency to hold on to the ball was visibly annoying his team-mates, was denied another penalty after being sandwiched by two Auchinleck defenders and Skacel had an effort saved by Leishman soon after.
Milliken forced a save out of Hearts goalkeeper Jamie MacDonald after 40 minutes but should have done better when meeting a cross six yards out.
Hearts had further chances but exited for the break with cheers from the visiting support outdoing jeers from the home fans.
Sergio made two changes at the break, Templeton replacing Novikovas and left-back Danny Grainger, who was making his return following a groin injury, replaced by Jordan Morton.
Driver shot wide and Templeton wildly over as the second half began in much the same pattern as the first.
Barr landed heavily after 63 minutes when attempting to meet a corner and, after a lengthy delay, exited on a stretcher.
Leishman made a point-blank save one-on-one from Skacel after 75 minutes as again Auchinleck survived.
A mix-up in the box then allowed Hearts to score.
Leishman crashed into substitute Bryan Slavin trying to meet Morton's long ball and it broke for Smith to drive low into the net.
Leishman saved well from Templeton but Hearts advanced to the fifth round - but only after Pope's effort was chalked off.