The 1-0 win, courtesy of Nicky Cadden's penalty, kept Livi a point clear of Airdrieonians, who have a game in hand and beat Peterhead 4-2 away.
Queen's Park moved above Albion Rovers into fifth place by beating Brechin.
The Spiders, who won 2-0, are now five points behind third-placed Brechin, with Alloa, who are fourth, inactive.
The Wasps, like Stranraer, East Fife and Stenhousemuir, were not in league action, though Stenny beat Forres Mechanics in the Scottish Cup.
At Cliftonhill, Ross Dunlop fouled Dale Carrick in the box and Cadden converted the spot-kick in the penultimate minute.
There was further misery for the hosts as Alan Reid was sent off in stoppage time following his second booking.
Airdrie raced into a three-goal lead at Balmoor with Andy Ryan's strike, Sean McIntosh's header and Iain Russell's finish all coming within the opening 20 minutes.
Nicky Riley replied for the hosts before half-time and Leighton McIntosh's penalty following Jack Leitch's foul on Riley further reduced Peterhead's arrears.
But Ryan's second effort of the match deep into stoppage time clinched the visitors' win.
Brechin City lost ground on the top two after losing at Hampden.
Ross Millen opened the scoring from the penalty spot after a first-half foul by goalkeeper Graeme Smith on Dario Zanatta and Ryan McGeever headed home a late Sean Burns corner.

World number three McIlroy has missed the cut at his home event for the last three years and has recorded just two top-10 finishes in nine attempts.
"My performances haven't been what I'd want but it's a tournament I would love to win one day," said the 27-year-old.
"I feel like my game is in good enough shape to do that."
The Northern Irishman, who is tournament host for the second year, recorded his best result in 2008, when he finished seventh in his first full year as a professional.
That came two years after he watched the final day of the 2006 Ryder Cup on the course as a 17-year-old spectator.
But, despite being frustrated at failing to turn good performances into a victory so far in 2016, McIlroy thinks he can contend for the title.
"The Irish Open always was, but even more so now, one of the most important weeks of the year for me," said the four-time major winner.
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"I'm coming off a couple of decent weeks in the States where I felt I played better than the results suggested, and the K Club is a course I feel I can do well on.
"It would be huge [to win]. Anyone that plays professional golf, they dream of winning their home open. You don't get many opportunities so it would be very special. It's one tournament that is missing from my CV."
Asked if he could pinpoint why he had not performed well in Ireland, McIlroy said: "No, not really. Maybe the pressure of playing at home - we don't play at home often - and maybe trying too hard or putting too much pressure on myself.
"There were obviously reasons I wanted to get involved in this tournament. I wanted to contribute something, where coming to the Irish Open was becoming more of a burden instead of something I relished and enjoyed.
"Being able to get involved and not just play for myself, but play for and help other people, I enjoy that part of it.
"I want to help out. I want to make this one of the best tournaments in the world and we've made a few good strides towards doing that."
McIlroy believes he learnt a lot from acting as tournament host last year, when he slumped to an opening round of 80 and missed the cut at Royal County Down, but was on hand to present the trophy to eventual champion Soren Kjeldsen.
"There were times where I spread myself too thin last year with a lot of commitments and we've tried to scale that back this year," he added.
Livingston maintained their slender advantage at the top of Scottish League One with a narrow and late victory over Albion Rovers.
[NEXT_CONCEPT]
Rory McIlroy is confident he can improve on his poor record in the Irish Open when he competes at this week's event at the K Club in County Kildare.