Ralf Rangnick's side were indebted to David de Gea for a superb first-half display that kept Brentford at bay before United finally sprang to life in west London.
Anthony Elanga put United in front early in the second half before Mason Greenwood and Marcus Rashford wrapped up the points.
United are the first club to record 300 Premier League away victories despite Brentford's late reply from Ivan Toney.
They sit seventh in the table, two points adrift of the top four as they chase qualification for next season's Champions League.
