A second-half penalty from Jordan White - his third goal in two games - pulled the visitors level and they could have won it in the closing stages.
Zander Clark produced fine saves to keep out Burton O'Brien and Gary Glen late on. By that stage Queens were down to 10 men after James Fowler was sent off for picking up his second booking.
The home side controlled the first half and Gavin Reilly missed a good chance early on from the edge of the six-yard box. Jon McShane went close with a shot from outside the penalty area and it was no surprise when they took the lead via an unfortunate own-goal from Kyle Jacobs.
Livingston manager John McGlynn changed it at half-time and his team improved. Glen missed a good chance to equalise but his composure deserted him in front of goal.
Livingston equalised courtesy of White's 65th-minute spot kick and six minutes later Fowler was red-carded.
Alloa Athletic gave Cowdenbeath their first home league defeat since January and did so with room to spare, winning 3-0 at Central Park.
Things started to go wrong for Cowdenbeath early on when Craig Sutherland went off injured before his replacement, Craig Johnston, also limped off after only 16 minutes.
Alloa took the lead seven minutes from the break when Kevin Cawley scored his first of the season. Greig Spence doubled their advantage three minutes into the second half and Liam Buchanan killed off Cowdenbeath with a third after 54 minutes.