Jose Mourinho had said his team would "probably lose" because their current fixture congestion would lead to fatigue, but despite missing Zlatan Ibrahimovic, Paul Pogba, Wayne Rooney and Ander Herrera, they counter-attacked well.
Belgium international Marouane Fellaini headed them in front after almost half an hour's play for his first league goal of the season and Jesse Lingard scored the second just after the hour mark from 20 metres.
Boro' substitute Rudy Gestede pulled a goal back with 12 minutes left, but in added time goalkeeper Victor Valdes slipped, allowing Antonio Valencia to score a third United goal.
Middlesbrough, by far the lowest scoring team in the league, have still not won a league game in 2017.
With United going into the game unbeaten in the league since October, there could hardly have been a greater contrast between the two sides' form.
By halftime Mourinho's side had made twice as many chances and taken the lead.
Valdes, after saving with his legs in quick succession, from Rashford and Valencia, was caught in no-man's land when Ashley Young crossed for Fellaini to head home.
It was the Belgian's first in the league since December 2015.
The second goal came from a classic break after regaining possession. The home defence stood off for too long, allowing Lingard to run on and score with a fierce drive from just outside the penalty area.
Boro' did show greater attacking intent than in many games this season and pulled a goal back when Chris Smalling failed to clear and Gestede poked in from close range.
Any hopes of a dramatic recovery were dashed, however, when Valdes fell over lining up a clearance and Valencia was able to walk the ball into an empty net.
Mourinho praised the "attitude and desire" of his players.
"Our guys gave absolutely everything and were magnificent," he told BT Sport.
Victory made United the first club to record 600 wins since the Premier League began in 1992.
They moved to within three points of fourth-placed Liverpool while Boro, who have scored only three league goals this year and 20 all season, remain bottom but one.
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