Marcus Rashford's first-half free-kick was superbly saved by Illan Meslier and Bruno Fernandes dragged his shot from in front of goal wide in the second period as the visitors went close.
But Leeds also passed up chances. Helder Costa's effort was deflected over the crossbar by Aaron Wan-Bissaka, and substitute Mateusz Klich shot straight at Dean Henderson when well-placed to score.
Solskjaer's side, 10 points behind City with five games to play, extended their unbeaten Premier League away run to 24 matches and are now undefeated in their last 13.
The Norwegian was delighted with his players' physical effort against a Leeds side renowned for their endurance.
