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.
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.
"Today we just didn't have the moment, take (our chances) when we had them," Solskjaer said.
"A draw is disappointing because we wanted to put pressure on (Manchester) City, but we couldn't 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.
"I've not seen many teams in the second half dominate as much as we did against Leeds, because they are so fit and strong," Solskjaer said.
"The intensity of their game has been so good, they steamroll teams, but I felt we almost did that to them second half.
"I really liked what I saw out there, we dominated and played in their half.
"We kept them away from our goal, away from our half really, so I'm really happy about that."
Solskjaer said none of his players or staff had seen the banner flown above Elland Road before kick-off, which read: "£2bn stolen #Glazersout".
The fall-out from the attempted European Super League coup would not be a distraction for his side between now and the end of the season, he said.
"I'm really happy with the boys' focus," he added. "They have such a determination to finish the season strong.
"So we're very focused to do well against Roma (in the Europa League) and then we've got Liverpool in between, so I'm not worried about that at all."
Marcelo Bielsa, whose side have not beaten their arch rivals in the top flight since 2002 when Solskjaer was a Manchester United player, was equally as pleased with his players' display.
"It was a very demanding game for us," Bielsa said. "For the game not to become unbalanced, the team made a massive effort.
"This effort allowed the game not to become unbalanced for us. The game for them came more naturally to them.
"Even if they dominated in the main, we managed the chances on goal similarly and even if they did have more of the ball to attack and they had a lot more attacks than we did, I insist I value everything we did so the game would not become unbalanced."
Leeds, who climbed above Arsenal into ninth place, are now undefeated in six Premier League games for the first time since 2002.
That run has also included draws against 'big six' sides Chelsea and Liverpool and victory at Manchester City.
Bielsa added: "What was touching and moving was the effort the team made.
"As the game grew, the spaces opened up and (with) more clarity to think we could go and attack. A draw was good enough reward. A sufficient result."