Tottenham is just a point behind second-place United after beating Aston Villa 2-0 for a fourth straight win, but the misery continues for Liverpool courtesy of a frustrating 0-0 home draw with Stoke.
Jonny Evans and Patrice Evra scored headers from corners for United in the opening 15 minutes and could then celebrate a rare clean sheet this season after Cleverley's long-range curler in the 71st made the game safe at St. James' Park.
"We came here, defended brilliantly. We fully deserved the victory," said England striker Wayne Rooney, who starred in his role as an attacking midfielder for United. "The first six games of the season we haven't defended well enough — not just the back four, the whole team, and today we were brilliant against a physical team."
United is level on points with Manchester City and four behind Chelsea after seven games, with the standings starting to take shape and the likely title contenders back among the frontrunners.
Tottenham can be included among them, with goals by Steven Caulker and Aaron Lennon moving the team to fifth — level on points with fourth-place Everton.
Spurs manager Andre Villas-Boas dropped goalkeeper Brad Friedel to the bench to end the American veteran's record run of 310 Premier League starts in a row, dating back to May 2004. His replacement, summer signing Hugo Lloris, barely had a save to make at White Hart Lane on his league debut.
"Obviously Brad was disappointed but as a great professional, he ended up accepting the decision," Villas-Boas said. "We respect his performances and I'm sure he will be back in the team as soon as possible."
Liverpool remains languishing in the wrong half of the table after failing to break down a stubborn, physical Stoke backline at Anfield, although the hosts struck the post twice through Raheem Sterling and Martin Skrtel.
"We just needed the final touch in the box to round up some good play," said Liverpool manager Brendan Rodgers, whose team is in 14th place following its worst start to a league season in more than a century.
In Sunday's other match, Jose Fonte scored a late equalizer — his second goal of the game — to earn Southampton a 2-2 draw against Fulham and lift the team out of the bottom three.
United lost 3-0 at Newcastle last season and there were fears the hosts' stellar frontline of Demba Ba and Papiss Cisse would cause havoc against a defence that has been criticized heavily in recent weeks.
Yet for all United's big names in attack — Robin van Persie and Danny Welbeck also played alongside Rooney — it was its defence that guided the team to victory and only a second shutout in the league.
Evans powered home a header from Van Persie's centre in the eighth minute before Evra got in front of Ba to glance in Rooney's inswinging corner.
"Unfortunately, with all talent they have on the pitch, it was set plays they scored from," Newcastle manager Alan Pardew said. "The two goals killed us."
United goalkeeper David de Gea clawed away a stooping effort from Cisse with the ball inches from crossing the line, moments after Ba headed against the crossbar, before Cleverley sealed the three points with a curling shot from the corner of the area.
At first, it appeared Cleverley meant it as a cross — and that's what United manager Alex Ferguson thought — but the England midfielder insisted he had gone for a shot.
"I meant to put it in the far corner," he said, with a cheeky grin.
The pressure was on United after seeing Chelsea and City romp to easy wins on Saturday and given it had lost 3-2 to Spurs in its last league outing. A victory was also important as teams now have two weeks off for the international break.
"We were focused today," Ferguson said. "It's not easy coming to Newcastle; they're a big, aggressive team. I'm pleased with the result and pleased with the performance."
After a slow start to the season, Tottenham is in fine form and was a deserved winner over Villa, going ahead when Caulker diverted in a shot from Jermain Defoe that was heading wide in the 58th.
Lennon then shot past Villa goalkeeper Brad Guzan in the 67th after being set up by Clint Dempsey.
"Overall a great performance," said Villas-Boas. "We told the players at halftime to persist with the intensity and we would come good."