Elsewhere, Tottenham Hotspur chalked up their fourth successive league win by beating Aston Villa 2-0 with goals from Steven Caulker and Aaron Lennon.
Liverpool's struggles continued when they were held to a goalless draw by Stoke City at Anfield and promoted Southampton needed a last-gasp goal from defender Jose Fonte, his second of the game, to rescue a 2-2 draw with Fulham.
Manchester United, desperate to bounce back from last weekend's surprise 3-2 home defeat by Spurs, made a high-tempo start at Newcastle and were rewarded in the eighth minute when defender Jonny Evans scored with a towering header from a corner.
"The way we played in that first 20 minutes was excellent," Manchester United manager Sir Alex Ferguson told Sky Sports television.
"It was strange in a way because we played some good football and made a lot of good openings but those two goals both came from set pieces."
Demba Ba shook the post with a header before goalkeeper David de Gea clawed away Papiss Cisse's follow-up attempt with the ball perilously close to going over the line.
"I think it was inconclusive," said Ferguson. "But I would have been disappointed if that had not been given to my team because most of the ball looked over the line."
Cleverley capped a sparkling show from Manchester United by curling a 25-metre shot into the top corner of Harper's net in the 71st minute.
"I think it was a cross but Tom said it was a shot," added the Manchester United manager.
The win lifted Ferguson's team up to second in the table, level on 15 points with Manchester City. Chelsea are top on 19 after sweeping Norwich City aside 4-1 on Saturday.
Spurs, unbeaten in all competitions since the opening day of the season, broke through in the 58th minute at home to Villa when Jermain Defoe's shot was deflected into the corner of the net by Caulker.
Earlier, Spurs manager Andre Villas-Boas decided to end Brad Friedel's record run of consecutive Premier League appearances by bringing in France goa
