Craig Levein's side had mustered just five goals in their first three Group A games despite fashioning 75 openings.
The Tynecastle boss ordered his side in for extra target practice this week but the Jambos were no sharper as they came up against Darren Young's Fifers.
Jamie Walker swept the Gorgie men ahead early on but their lack of lethal edge returned to haunt Levein's team as Chris Duggan levelled for the League One part-timers nine minutes into the second half.
They even missed out on the bonus point and a seeding for the second round as East Fife won the shoot-out 7-6 to pinch the runners-up slot from Dundee United after Sean Clare, Oliver Bozanic and Aaron Hickey all failed to convert.
Levein had laid on a number of crossing-and-finishing drills in a bid to help his team rediscover their killer touch, but it is unlikely he had Craig Halkett in mind for the role of provider.
Yet it was the former Livingston centre-back who teed up the opener 15 minutes in as he somehow found himself wide on the right, where he robbed Pat Slattery before slinging a low ball to the back post for Walker to stroke home.
Uche Ikpeazu passed up a couple of chances as again the Jambos failed to embellish their advantage and that only offered encouragement to the hosts.
There was a warning just before the break when Scott Agnew's free-kick from wide broke to Duggan. His hooked volley forced Zdenek Zlamal to backpedal before pushing over the crossbar.
But it was not heeded and Hearts paid the price after 54 minutes as Ryan Wallace's cross from the right found Duggan again, with the Australia-born frontman guiding a superb finish into the top corner.
Oliver Bozanic had a decent free-kick palmed away by Long late on but the travelling support let rip at full-time as their laboured display failed to bring a winner.
And the boos aimed at Levein continued as Aaron Dunsmore fired home East Fife's winning penalty.