United managed just two shots on target all game while Argyle had only a solitary effort that tested Neil Sullivan.
Kevin Blackwell brought Frazer Richardson and David Healy into the starting line-up as he looked to freshen up a side that had not scored in the last three games.
But it made little difference during a dour first half in which neither side managed a single shot on target.
Healy's square ball gave Gary Kelly room to shoot from 20 yards in only the second minute but his effort flew over and it was to signal the standard of things to come.
Argyle skipper Paul Wotton repeated the trick at the other end 10 minutes later as both sides struggled to create a clear chance.
When that chance eventually came, a mix-up between Robbie Blake and Rob Hulse saw Hulse pull his shot wide when he would have been better returning the ball to Blake.
Leeds were forced into a reshuffle midway through the first half when Kelly, who had been suffering with flu all week, was withdrawn and replaced by rookie Jermaine Beckford.
But even with four strikers on the pitch, Leeds were still only average at best with Healy's curling strike rebounding off the post in the half's only bright spot.
Tony Capaldi and David Norris sliced open the Leeds defence with 25 minutes remaining but Sullivan reacted well to keep out Leon Clarke's well struck effort.
Hulse could have given the hosts a vital lead soon after when he latched on to Beckford's clever flick only to see the ball bobble away from him as he was bearing down on Plymouth keeper Romain Larrieu.
Beckford finally managed to get Leeds' first shot on target in the 80th minute but Larrieu came to the rescue before also stopping Blake's free-kick as Blackwell's men were forced to settle for a point.