Rooney converted from close range with 16 minutes left after good work by substitute Ji-sung Park to ensure the Reds extended their lead over Chelsea at the top of the table to three points.
Cristiano Ronaldo’s 37th goal of the season had given United an early lead, but Alves produced two superb finishes in each half to put Boro in command in an end-to-end game.
United’s defence, shorn of Nemanja Vidic, was frequently made to look uncharacteristically shaky by a lively Boro outfit and only a series of last-ditch tackles prevented the Teessiders from extending their lead in the second half.
Rio Ferdinand came off with what looked like a leg injury with 20 minutes left and, although United also had chances to win it, Sir Alex Ferguson will perhaps settle for a point after a testing encounter in the driving north-east snow.
As so often this season, United began brightly and took the lead inside 10 minutes. Ryan Giggs, making his 750th United appearance, swung over a deep corner to Michael Carrick, who made space before firing the ball back across goal for Ronaldo to convert left-footed from close range.
It was the winger's 27th league goal of an extraordinary campaign and it looked set to give United the platform to go on and dominate. Indeed, had the Reds taken one of several more chances in the opening 20