There was no hint of a giant-killing at Tynecastle as Jim Jefferies’ charges snuffed out the visitors’ threat with two first-half strikes.
Eggert Jonsson provided the 11th-minute breakthrough before Kevin Kyle scored his first Tynecastle goal soon after the half-hour mark.
Arvydas Novikovas, the Lithuanian midfielder, claimed an impressive third before substitute Scott Robinson wrapped it up late on.
Jefferies rang the changes for the Moray men’s visit, with Kyle – making his first start – Jonsson, Novikovas, Ryan Stevenson and Janos Balogh brought in after Saturday’s 4-0 win at Hamilton.
Elgin, bottom of the Third Division after three defeats and the concession of 13 goals, appeared in for a difficult night.
But with veteran winger Barry Wilson in their ranks they at least had valuable experience and started the contest brightly.
Yet it took only 11 minutes for the Clydesdale Bank Premier League outfit to break the deadlock.
Craig Thomson’s corner was not properly cleared and Ian Black’s lofted ball back into the danger area saw Jonsson net, if a little luckily as the ball spooned up and over Steven Dunn.
It was to soon prove one-way traffic thereafter with Novikovas curling just wide and then Kyle denied at the back-post by a smart Dunn save.
Hearts were peppering the Elgin goal and only an incredible save from the right boot of Dunn prevented Black slotting home from David Templeton’s cross.
The visiting custodian was already having a busy evening and endearing himself to the noisy band of away fans with some fine stops.
He next threw himself at a Stevenson header before Ruben Palazuelos caught the post with the rebound.
Jason Crooks broke upfield to give his defenders a breather, the striker firing wide from the right to register City’s first shot on goal on the half-hour mark.
But it was to prove a brief highlight for Ross Jack’s men as Hearts doubled their lead in trademark style from Kyle on 33 minutes.
Another fine right-wing centre from Thomson saw the striker bullet home a header.
Hearts were in no mood to take their foot off the accelerator in the second half and Novikovas finished a flowing move by netting a deflected 18-yard drive just short of the hour.
There was almost consolation for Elgin soon after but Balogh got down at his near post to block out Craig Gunn’s close-range chance.
Six minutes from time substitute Robinson added further gloss for Hearts by clipping in a fourth.