Kilmarnock had to work hard against 10-man East Fife to progress to the semi-finals of the Scottish League Cup.
An evenly-contested first half ended with Robert Ogleby being sent off after picking up a second yellow card.
Killie made the breakthrough on 73 minutes when Mohamadou Sissoko latched onto a spilled ball by Fifers goalkeeper Marc Ridgers.
The tie was over 10 minutes from time when Gary Harkins pounced on another loose ball to blast home from 14 yards.
East Fife had taken care of Dunfermline and Aberdeen earlier in the tournament and gave the hosts a few first-half scares, with Ogleby their main threat.
Kilmarnock dominated possession for long spells and leading scorer Paul Heffernan turned and shot wide after a well-worked early corner.
Ogleby then replied with two dangerous shots as he showed Killie skipper Manuel Pascali a clean pair of heels on both occasions.
Ridgers made comfortable saves from each of Killie's front three players - Harkins, Dean Shiels and Heffernan and home goalkeeper Anssi Jaakkola was called into action just before the interval 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.
The Wales Under-21 striker had been booked for a reckless challenge on Danny Buijs and received his second yellow card when he caught Rory McKeown with a kick after the defender had dispossessed him.
Buijs fired just over as Killie stepped up the pressure against depleted opponents in the second half and Shiels was denied by a great last-ditch challenge from John Ovenstone, while Heffernan's 20-yard strike shaved the post.
But the 10 men were not consigned to defending and Sissoko to divert needed to divert a shot from Ryan Wallace over and David Whyte could not turn Jon Ovenstone's flick on target from close range from the resulting corner.
Kilmarnock substitute David Silva saw a long-range effort fizz over the crossbar before a goalkeeping error presented the hosts with the opening goal.
Ridgers, who saved three penalties in the shoot-out with Aberdeen, came for a 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 by a good tackle just inside the box.
Kilmarnock manager Kenny Shiels: "When I first came to the club, they told me it was the second biggest derby in Europe after Barcelona - Ayr against Kilmarnock.
"It's mouth-watering prospect that one, I'm told, because there is an intensity there.
"Even the little villages outside Kilmarnock, I'm told some are all Ayr United and some villages are all Kilmarnock supporters.
"I don't really have a view because you have to respect the other teams, but Kilmarnock and Ayr would be a good draw.
"I'm delighted for my chairman and the club.
"To be in the semi-finals of the cup and one point off the top six, I think is a fantastic start for the club when you consider we've had to rebuild a team within a short period of time.
"I'm pleased to get to the semis because it was nervy out there at times."
East Fife manager John Robertson: "I'm disappointed for the players and the fans because I genuinely believe, if it had stayed 11 against 11, we could have been sitting in the semi-finals of the Cup.
"We felt really disappointed to come in level at half-time - and a man down. They played some lovely football, but they didn't hurt us.
"When you talk about cup upsets, you need to be at your best, which we were.
"You need to stop the opposition playing, which we did magnificently well. And you need that wee ounce of luck and I think that's the only thing that let us down."