Second-placed Brechin City and Alloa Athletic, who are third, also won against Peterhead and Stenhousemuir respectively.
Airdrieonians, who occupy the other promotion play-off place, beat Albion Rovers 2-1 to stretch their advantage over their opponents to five points.
And Queen's Park were 2-0 winners away to Stranraer.
At Livi, Liam Buchanan's strike in first-half stoppage time put the hosts ahead and Michael Miller's free-kick doubled their advantage before Kevin Smith netted to halve the visitors' deficit.
Mark Hurst was dismissed for a foul on Danny Mullen and Buchanan scored the resulting penalty before the visitors also had Paul McManus sent off.
Ross Caldwell scored a hat-trick as Brechin won 3-1 at Balmoor, Rory McAllister netting the hosts' consolation.
In Clackmannanshire, Alloa comfortably overcame bottom club Stenny, with Greig Spence's strike and Andrew Graham and Jamie Longworth's headers putting the Wasps in command.
Alistair Roy replied for the visitors but Kevin Cawley rounded off a 4-1 win to keep Alloa four points behind Livi.
Airdrie went ahead at Cliftonhill when Andy Ryan lashed into the top corner and Iain Russell doubled their advantage, Ross Stewart heading one back for Albion Rovers.
And goals by Dario Zanatta and Anton Brady secured the three points for the Spiders at Stair Park.
Livingston remain two points clear at the top of Scottish League One after they beat nine-man East Fife.