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The point means that Liverpool, who have 11 games remaining, will be disappointed not to have taken advantage of an injury-ravaged United.
Alvaro Morata's second-half brace was enough for Chelsea, who beat United in last year's final, to move through.
Manchester United were 11 points off the fourth Champions League spot when Solskjaer took over a fortnight ago but their gap behind Chelsea is now already down to six points.
The re-born Paul Pogba scored twice, for the second time since Jose Mourinho's departure from the club, as Manchester United beat Bournemouth 4-1.
Unbeaten Liverpool have 45 points from 17 games with champions Manchester City, who beat Everton 3-1 on Saturday, a point behind.
Chelsea's win moved them temporarily into third on 34 points, ahead of Arsenal on goal difference.
Liverpool also fell behind but stormed back to beat Burnley 3-1 and cut the gap on leaders Manchester City to two points.
Haris Seferovic scored a hat-trick as Switzerland recovered from conceding two early goals to beat World Cup semi-finalists Belgium 5-2 on Sunday.
Eighth placed Manchester United are now 12 points behind City.
United are in provisional seventh spot on 20 points from 11 games.