Juan Mata curled in a second-half free kick after Marouane Fellaini's equalizer to give United a 2-1 victory, keeping the 20-time champions in fourth place.
Liverpool, which edged Stoke 1-0 at the weekend, came from behind to beat 10-man Leicester 3-1. After an own goal by Liverpool goalkeeper Simon Mignolet, Adam Lallana equalized before Gerrard and Jordan Henderson clinched Liverpool's second successive win in the second half.
Liverpool, though, remains five points from fourth-place United in eighth. West Ham stays fifth after fighting back to edge West Brom 2-1, and Swansea is a place behind after beating Queens Park Rangers 2-0.
QPR is only being kept off the bottom by Leicester, while Burnley provisionally climbed out of the relegation zone by drawing 1-1 with Newcastle. Aston Villa claimed its first league victory since September by winning 1-0 at Crystal Palace, with Christian Benteke scoring his first goal in nine months.
The attention returns to the title race on Wednesday when leader Chelsea hosts Tottenham, second-place Manchester City plays at Sunderland and third-place Southampton is at Arsenal.
But Manchester United, by keeping up its resurgent form, is showing it deserves a place back among the elite.
Without the injured Wayne Rooney and Angel Di Maria, United lacked creativity at Old Trafford. But Fellaini met Ander Herrera's cross with a towering header to put the hosts in front in the 21st minute.
Steven Nzonzi equalized in the 39th for Stoke with a powerful shot from the edge of the area, but United's lead was restored 20 minutes later from Mata's inswinging free kick.
United needed a superb save from goalkeeper David de Gea and a goal-line clearance from Ashley Young in the final two minutes of stoppage time to preserve the victory.
"I was not pleased with the first half," United manager Louis van Gaal said. "In the second half we played much better, kept the positions."
Liverpool entered the game at Leicester with uncertainty hanging over Gerrard's future after the 34-year-old midfielder was dropped from Saturday's lineup, and with a new contract yet to be signed for next season.
Gerrard showed his enduring value to the team by powering in Liverpool's decisive second goal in the 54th minute, but remained coy about his future.
"I'll decide when I'm ready," the former England captain said. "There's nothing to say on the contract at the moment."
Liverpool's fightback ensured the team provisionally rose to eighth — far from matching last season's runners-up finish.
Liverpool fell behind in the 22nd when Leonardo Ulloa's shot hit the post and rebounded into the net off goalkeeper Simon Mignolet.
But Adam Lallana levelled four minutes later and — after Gerrard's goal — Leicester had Wes Morgan sent off for dragging down Rickie Lambert. With a man advantage, Jordan Henderson completed Liverpool's victory in the 83rd.Suggest a correction