Jurgen Klopp's side would have gone top with a victory against their bitter rivals at Old Trafford.
But despite taking the lead through Luis Diaz's 23rd minute strike, Liverpool couldn't deliver the knockout blow.
They wasted several chances to increase their advantage before Bruno Fernandes alertly pounced on Jarell Quansah's wayward pass to score from just inside the Liverpool half in the 50th minute.
To Klopp's visible fury, Kobbie Mainoo found space to net United's second with an eye-catching curler in the 67th minute.
Liverpool avoided a repeat of their FA Cup quarter-final defeat at United earlier this season as Mohamed Salah equalised with an 84th minute penalty after Aaron Wan-Bissaka chopped down Harvey Elliott.
But it was still a frustrating final visit to Old Trafford for Klopp before his departure at the end of the season, with Liverpool stuck behind Arsenal, who hold a nine-goal advantage on goal difference.
Klopp's team, who are one point ahead of third placed Manchester City, have seven games left to overhaul Arsenal as they chase a record-equalling 20th English title.
"We should have won the game, that is clear. We should have had other goals in the first half. We dropped too deep," Klopp said.
"The whole Liverpool community just stay calm. We are where we are and we keep giving it a proper go.
"Do I wish we had 10 points difference? Of course. We are in the race and I'm absolutely fine with it."
United boss Erik ten Hag added: "Very mixed emotions. We have to blame ourselves for making stupid mistakes.
"On the other side I'm very proud. You see how we are improving and the potential of this squad is amazing."
Slick Spurs
At the Tottenham Hotspur Stadium, the game went ahead despite a fatal stabbing just yards from the ground earlier on Sunday.
Ange Postecoglou's side took the lead thanks to an own goal from Forest defender Murillo in the 15th minute.
Timo Werner's low cross into the six-yard box caused panic and Murillo's stretching attempt to clear diverted the ball into his own net.
Chris Wood equalised in the 27th minute with a clinical strike from Anthony Elanga's pass for his 12th league goal this season.
But Micky van de Ven bagged Tottenham's second in the 52nd minute with a blistering strike from just inside the area.
And six minutes later Pedro Porro produced a powerful half-volley from 14 yards to put the result beyond doubt.
Tottenham are now above fifth placed Aston Villa on goal difference and also hold a game in hand in the race to qualify for the Champions League via a top four finish.
"Everyone has been banging on about us getting fourth. We're fourth now but it doesn't stop, we'll keep going. We're in a good shape to finish the season strong," Postecoglou said.
Fourth bottom Forest are outside the relegation zone by the slenderest of margins, above Luton only on goal difference.
In Sunday' other game, Ole McBurnie's last-gasp equaliser forced Chelsea to settle for a 2-2 draw at bottom of the table Sheffield United.
Fresh from scoring twice in the final seconds of stoppage-time to beat Manchester United on Thursday, Chelsea took the lead after 11 minutes as Thiago Silva finished off Conor Gallagher's corner at the far post.
The Blades equalised in the 32nd minute as Gustavo Hamer's pass sent Jayden Bogle through on goal and the right wing-back slotted home.
Noni Madueke put Chelsea back in front in the 66th minute, cutting in from the right to drive his shot into the far corner.
Yet mid-table Chelsea's troubled season suffered another setback when McBurnie levelled with a close-range finish three minutes into stoppage-time.
"We feel really bad in the way we conceded. We need to be more clinical," Chelsea boss Mauricio Pochettino said.
