The Red Devils remain in the lower half of the table after the result at Old Trafford.
But Solskjaer was optimistic for the future after seeing his side almost beat the league leaders.
"A great response from the boys. We know we have a team who work for each other and there's a great atmosphere," Solskjaer told Sky Sports.
"Maybe this will turn the season. They're disappointed in the dressing room because they know they should have won.
"That's a really good sign for a manager, to see your team disappointed not to beat Liverpool. Our fans can see what's happening. We will get there.
"It's important to get results, it's the only way to grow confidence in what we're doing. A win would have been great but a draw is a step in the right direction.
"Newcastle [a 1-0 loss] was a low point for us. We've got players coming back from injury and illness and we'll only improve because more will come back very soon. I'm pleased.
"You can't look too far. You have to win games. We're Manchester United and need to win more games. It was a lost opportunity but we go to the next one."