It was a young Hibs side with the starting line-up composed almost entirely of under-18 players while the Moray outfit fielded more or less their first-choice XI.
Both teams took a while to settle into the game with Robbie Hamilton heading an Elgin corner clear on 20 minutes before Connor Young was flagged offside from a good position before he could pull the trigger.
Conor O'Keefe was the main attacking outlet for Elgin on the left and Josh McCulloch was kept busy by the former St Mirren winger in the opening stages.
Gregor Fordyce hacked an effort off the line as the League Two side sought the opener and on the half-hour mark Kane Hester headed straight at Murray Johnson from close range when it looked easier to score.
But it was Hester who eventually opened the scoring, converting from close range after Conor O’Keefe had drawn Johnson from his goal-line and fed the ball into the six-yard area.
There was more than a degree of controversy surrounding the opener; the Hibs players and staff remonstrating with match referee Calum Scott after what looked like a foul on Innes Murray in the build-up had gone unpunished, while there was more than a suspicion of offside when Hester put the ball in the net.
McCulloch and Fordyce both had efforts from distance in the moments following the goal while Hamilton shot narrowly wide from a tight angle. Josh O'Connor continued to cause the visiting defence problems and one surging run took him to the byline and allowed him to cut the ball back for Murray Aiken, whose first-time effort was dealt with by Tom McHale in the Elgin goal.
Darryl McHardy then had a gilt-edged chance to double the SPFL side's lead but he contrived to head wide of Johnson's right-hand post from a corner.
Two minutes before the break McHale pulled off a fantastic save to deny O'Connor the equaliser, tipping the ball over the bar at full-stretch before thwarting Aiken and McCulloch in quick succession.
O'Keefe then went close again for Gavin Price's side but although he beat Johnson his effort flashed wide of the far post.
The visitors increased their lead four minutes after half-time, Hester grabbing his second of the evening as he latched onto a cross-field ball and held off Jacob Blaney's challenge before lifting it over the onrushing Johnson.
Aiken went close for Hibs on the hour mark while Johnson had to be alert to deny O’Keefe from distance. Former Hibs youngster Josh Peters then found the side-netting as Elgin sought a killer third while a Young free-kick after a foul on O’Connor wasn’t far away from McHale’s right-hand post.
Gareth Evans and Eddie May brought on Ethan Laidlaw for Fordyce with 25 minutes remaining in a bid to change up Hibs' fortunes and they were handed a golden opportunity to reduce the deficit after Creag Little gave away a penalty.
Young stepped up and struck the spot-kick well enough but although he sent McHale the wrong way his effort smacked the base of the post and went out for a goal-kick.
With 15 minutes remaining McCulloch cleared an Elgin effort off the line before Hester, looking for his hat-trick, sent an effort just wide.
Aiken gave Hibs hope in the dying moments when he bundled home after a stramash in the box but it was too little, too late with Elgin advancing to face Dunfermline Athletic in the next round.
The wee Hibees can be proud of their efforts against an established SPFL side, with a number of 16- and 17-year-olds giving a good account of themselves against players ten to 12 years older.
Hibs team: Murray Johnson; Josh McCulloch, Jacob Blaney, Jayden Fairley, Oscar MacIntyre; Innes Murray, Gregor Fordyce (Ethan Laidlaw 64); Murray Aiken, Robbie Hamilton, Connor Young; Josh O'Connor.
By Patrick McPartlin at Edinburgh Evening News.