The home side started brightly and Marc McNulty had a shot saved by Alan Combe. Morton went even closer and were only denied by Andy McNeil’s superb reflex save after a flick on by Andy Graham.
Combe matched that effort when he was at full stretch to divert McNulty’s shot for a corner.
The home defence then failed to deal with David O’Brien’s cross, but Livi were saved by Craig Barr’s athletic goal-line block.
Referee Des Roache found himself in the thick of the action on the stroke of half-time by denying a penalty claim when McNulty raced through on goal and appeared to be upended by Fouad Bachirou.
Combe dived bravely at the feet of Keaghan Jacobs on the restart, but the midfielder came off worse and had to be replaced by David Sinclair.
Livi enjoyed the bulk of the possession, but were unable to turn that pressure into goals.