The Norwegian saw his side take a first-half lead through Marcus Rashford as they looked to secure some respite from their worst league start to a season in 30 years.
But Jurgen Klopp’s team hit back with a late goal from Adam Lallana to preserve their unbeaten start to the campaign and stretch their lead on champions Manchester City to six points.
And former United boss Mourinho, who was replaced by Solskjaer in December of last year following a 3-1 defeat at Anfield, thinks the manager will rue the way his team let victory slip away from them late on.
"Liverpool did something in the last 20 minutes of the game to give Man Utd problems,” Mourinho told Sky Sports . “They got more people through the middle, more people in the box. But I think Ole and United, after such a great effort, will feel a bit disappointed.
"At the start of the game I was asking United for heart. They had it but they had more than that. They were organised, they had a gameplan that they stuck to. But [Graeme Souness] asked before the game if they could keep the defensive intensity for 90 minutes and they couldn't.
“Liverpool were on top at the end. The result is probably fair but if I were Ole I'd be proud but also frustrated."
The result leaves United still mired in the bottom half of the Premier League table and trailing Liverpool by 15 points, with the team’s well-documented problems seemingly further from being resolved than ever.
"I think both managers will be satisfied with the draw,” former United skipper Roy Keane said. “The pressure was building from Liverpool, and obviously their lads are talking quite critically about themselves, but I feel you have to give United a bit of credit today.
"They got the first goal and then were allowed to sit in but they got a little bit sloppy. United lacked a little bit of composure.
"With Liverpool, if you're talking about winning titles, you're talking about collecting points when you play badly, and that's exactly what they did."