With the home side short of attacking inspiration, Sissoko fired home in the 73rd minute after on-loan Hearts goalkeeper Mark Ridgers failed to collect a corner.
Another Tynecastle youngster, Robert Ogleby, had given the home side the initiative when he was sent off in first-half injury-time, before Kilmarnock finally took control with Harkins adding a second.
Wales Under-21 striker Ogleby had been causing problems for the Kilmarnock defence, notably Manuel Pascali, before receiving his second yellow card for a foul on Rory McKeown and had little cause for his complaints.
The Fifers, who had moved from bottom to fourth in the Irn-Bru Second Division since adding the scalp of Aberdeen to their earlier conquest against Dunfermline, had contained Killie well in the first half.
But the man disadvantage ultimately proved too much despite the continued frustration of their hosts for much of the second period.
Killie, who named the same team that drew with Celtic and Motherwell, created an early opening when Harkins worked a one-two with Paul Heffernan, but Ridgers turned the former Dundee player’s near-post effort wide.
Heffernan turned and shot wide after a well-worked corner and Killie continued to dominate possession, but the visitors were showing attacking intent whenever the opportunity arose and Ogleby fired just wide from a tight angle after bursting past Pascali.
Ogleby soon curled wide after getting in behind Pascali and the Italian was exposed again as Ryan Wallace broke clear, but he recovered to block and McKeown stopped the follow-up shot.
Ridgers made comfortable saves from each of Killie’s front three players – Harkins, Dean Shiels and Heffernan – but the Fifers were starting to frustrate the home fans while retaining a hint of attacking menace.
Shiels shot into the side netting following a Harkins pass but home goalkeeper Anssi Jaakkola was called into action just before the break to parry Robert Sloan’s 30-yard free-kick before full-back Matthew Park blazed the rebound over.
But the visitors suffered a serious blow when Ogelby was sent off by Mike Tumilty.
The striker had been booked for a reckless challenge on Danny Buijs and received his second yellow card after a poor tackle from behind, which left McKeown needing treatment.
The left-back did not reappear after half-time with his place taken by Garry Hay, who put in a series of dangerous crosses as Killie exerted some early pressure.
Buijs fired just over the top corner, Shiels was denied by a great last-ditch challenge from John Ovenstone and Heffernan’s 20-yard strike shaved the post.
But the 10 men, backed by a noisy support, were not consigned to defending. Wallace again got away from Pascali only for Sissoko to divert the shot over and David Whyte could not turn Ovenstone’s flick on target from close range after the resulting corner.
The East Fife defence continued to protect their goalkeeper, who saved comfortably from Harkins’ header before substitute David Silva shot just over from 25 yards.
The breakthrough owed more to a goalkeeping misjudgement than attacking guile.
Ridgers came for Hay’s corner but could not hold the ball amid a ruck of bodies and Sissoko fired into the top corner after controlling the loose ball.
The Scotland Under-21 squad member recovered to make a flying save from Shiels following a counter-attack as Kilmarnock looked to build on their lead.
Silva hit the post after an incisive move and Harkins doubled the advantage in the 80th minute when Ridgers could not keep out his angled drive after Heffernan’s progress had been halted just inside the box.