Hovering close to Lee Hollis's left post, the Stenhousemuir defender reacted just fractionally too late as Paul Jon Sludden's sliced shot skidded towards him, allowing Hearts to squeeze through in normal time.
After last week's decision to put out virtually a second XI for their Petrofac Cup tie at Livingston there was evidence that this competition figures a bit higher in Hearts' priorities, with four players retained from the side that thrashed Raith Rovers on Saturday.
They included James Keatings, who was doubtless hungry to get more goals after his hat-trick in Kirkcaldy. He got little opportunity to do so, however, in the 39 minutes he was on before picking up an injury and then hobbling off.
Even before that it had not looked like being Keatings' night, with the better chances falling to his strike partner, Gary Oliver.
He took the first of them when, unmarked in front of goal, he waited patiently for Liam Smith's cross from the left to reach him then did enough to direct it past Greg Fleming after only five minutes.
Five minutes later Oliver could have doubled his tally when Brad McKay's lofted ball took out the home team's defence but this time the Hearts forward could not direct his lob high enough to beat Fleming.
Midway through the half Oliver met Jamie Walker's deep ball beyond the far post but, while he flighted the ball over Fleming this time, he put just too much on the ball.
The visitors' rhythm was not disrupted by lengthy treatment to Keatings or the departure moments later of Jordan McGhee, who was replaced by Liam Gordon.
No sooner had the substitute come on than Hearts doubled their lead. Walker slipped the ball into the path of left back Kevin McHattie as he ran into the penalty area and, with a right-footed effort, he put it neatly to Fleming's left.
McHattie threatened again moments later when he fed Scott Robinson on the edge of the box and took the return in his stride, only for his left-footed strike to slide wide.
It looked as if Hearts were set to overrun their hosts but Stenhousemuir revived the contest with a 39th-minute goal.
Kieran Miller took his chance well, driving it low to Hollis's right after Martin Grehan had knocked Kris Faulds' cross down into his path 14 yards out .
If that re-energised them so did the changes made at half-time by Scott Booth, the home manager, as he introduced the lively Colin McMenamin and Alan Lithgow.
Fleming remained the busier of the keepers in a half that surprisingly went scoreless, given the opportunities created at both ends.
He made fine saves from Osman Sow - who showed good pace, power and footwork - Smith, Oliver and Walker. A Sow header also rebounded off the bar and Walker blasted arguably the best chance of the half over.
Hollis was also extended by a fine shot from Faulds after McKay had been caught in possession at the edge of his box. Then that late chance was spurned by Greacen.