The result moved Stranraer off the bottom of the division while Airdrie's visit to Brechin City was called off because of a waterlogged pitch.
Queen's Park moved into the top four with a 2-0 win over Stenhousemuir, who slipped to bottom on goal difference.
And Peterhead drew 1-1 with Albion Rovers in the other League One game.
Alloa Athletic v East Fife had previously been postponed due to the Fifers' Scottish Cup meeting with St Mirren. East Fife lost 3-2.
At Ochilview, on-loan Hearts forward Dario Zanatta fired Queen's Park ahead and although the Spiders had Ross Millen sent off for two bookings, Anton Brady added a stoppage-time second for Gus MacPherson's side. They are two points in front of Brechin, who have a game in hand, and a point off Alloa.
Eighth-placed Peterhead led Albion Rovers through Rory McAllister's cool first-half finish but Ross Stewart swept in an equaliser in the second half.
The two sides are level on points - four ahead of Stranraer and Stenhousemuir.
Livi, who threatened through Scott Pittman at home to Stranraer, are 11 points clear of Airdrie and 12 ahead of Alloa.
Scottish League One leaders Livingston were held to a 0-0 draw with Stranraer while nearest challengers Airdrieonians saw their game postponed.