During the closing stages Sir Alex Ferguson's men pushed frantically to extend their lead at the top of the table to five points with five games remaining, but they were met by stubborn resistance.
And Boro, who remain 14th, will be bitterly disappointed not to have emerged victorious having dominated for long spells and continually threatened a third goal.
United arrived knowing victory against Boro, followed by wins against Arsenal, Blackburn and Chelsea, would see them retain their Premier League title at Stamford Bridge on 26 April.
And, as if desperate to hasten their coronation, they made a blistering start and were rewarded within 10 minutes.
Boro have proved awkward opponents for United in recent seasons - winning five of their meetings since 2002.
But they seemed only too obliging as Emanuel Pogatetz missed Ryan Giggs's corner before Jeremie Aliadiere was beaten with ease by Carrick, who centred low for Ronaldo to guide home his 37th goal in all competitions and his seventh in six games.
Given that Boro had failed to score more than a single goal in their previous 16 matches, allied to the fact that United had not conceded in 461 minutes of league football, the likelihood of the hosts mounting a comeback looked slim.
But their response to going behind was emphatic and, while leaving themselves somewhat exposed at the back, Gareth Southgate's men attacked with craft and verve.
Ronaldo's 37th goal of the season gave United the lead
United's defence, stripped of the injured Nemanja Vidic, looked far from comfortable with the threat posed by Alves and Aliadiere, ably supported by Stewart Downing from the left and Gary O'Neill on the right.
On 17 minutes Alves rose above Patrice Evra but his connection with Andrew Taylor's cross from the left was weak and wide.
At the other end only a last ditch tackle by Luke Young prevented Rooney a clear sight of goal following slick build-up play between Ronaldo and Carrick, but the momentum was firmly with Boro.
With Evra lying injured from an O'Neill challenge, they were able to exploit the gap on United's left and Julio Arca sent Aliadiere racing clear, but the Frenchman scuffed his drive wide.
Ronaldo was never far from the action in a enthralling first half and signalled his omnipresent menace by beating four Boro players on the left only to slice his shot.
Alves, finally appearing to have grown accustomed to the pace of English football following his £12.5m move from Heerenveen in January, gradually began to assert more authority on proceedings.
The 27-year-old's swerving free-kick in the 32nd minute was parried by Edwin Van der Sar, who then repelled Pogatetz's follow-up header.
United would not be so fortunate two minutes later, though, as O'Neill's searing long-ball was cleverly flicked on by Aliadiere to the unmarked Alves, the Brazilian dinking a finish beyond Van der Sar.
Alves (right) had failed to score in his previous eight Boro games
Clearly believing they could add to their advantage before the break, Boro rallied once more and Downing was thwarted at the back post by an alert Van der Sar.
Although United felt they should have had a penalty - claiming a Ronaldo drive was blocked by Taylor's arm - it was Boro who began brightest after the break and Alves, collecting a David Wheater flick, saw his close-range header scrambled clear.
It was inconceivable to think Gareth Southgate's side are enduring their worst campaign in seven seasons, even more so when Alves put them ahead with a stunning finish.
Amid driving snowfall, Wes Brown could only nod George Boateng's high ball into the back of Aliadiere's head and it fell kindly for Alves, who opened his body and angled the ball into Van der Sar's far corner.
Alves came close to his hat-trick on 59 minutes when his header was cleared off the line by Rio Ferdinand after Aliadiere inexplicably cushioned O'Neill's cross back rather than going for goal himself.
Clearly incensed by his team's abject defending, Ferguson replaced Carlos Tevez with Park and John O'Shea with Owen Hargreaves, while 21-year-old Gerard Pique came on for Ferdinand, who limped off and will learn the extent of his injury on Monday.
While not necessarily pre-planned, the changes appeared to galvanise United and Park collected a Carrick pass on the right, broke past Taylor and squared for Rooney to score, albeit via a pivotal deflection off Wheater.
• Middlesbrough manager Gareth Southgate: "We were up against a team that are both outstanding to watch and to test ourselves against and we had them rattled.
"The team performance was excellent. We answered most of the questions and had them on the back foot for the majority of the game.
"We've now been to Chelsea, Arsenal, Aston Villa and given a good account of ourselves. They are a young group of lads who will grow as a team."
• Manchester United manager Sir Alex Ferguson: "We could have lost the game there's no question - they had some chances. At times the defending was a bit haphazard I thought.
"But in other ways we could have won the game because some of the approach and football was very good. At 1-0 we looked very strong.
"They changed their formation to a 4-4-2, got an equaliser, and it was a great game of football after that."
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