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.
Mourinho identifies United 'enemies' after losing Pogba
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.
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