Alves waited patiently for the fuss to die down and kept his cool to give Joe Hart no chance with the penalty.
Earlier in the season Middlesbrough manager Gareth Southgate had forecast that Alves would snap out of a lean scoring spell and demonstrate he was a goalscorer of international class.
The £12million signing has bounced back with two goals in successive games.
With memories of an 8-1 thrashing at the Riverside Stadium on the final day of last season, City were eager to avenge that humiliating scoreline.
Middlesbrough, however, hoped to redeem themselves in front of their own fans after a 5-0 rout by Chelsea in their last home game.
Robinho has set a personal target of 30 goals this season but he never really looked like improving on his six-goal tally.
He fired wide from outside the penalty area early in the game but in general he never posed a serious threat.
City skipper Richard Dunne came to his side's rescue when he challenged Alves for a high ball, but the central defender was injured in the process, although he quickly recovered.
Hart saved a shot on the turn from Tuncay but Ross Turnbull was the busier goalkeeper at the other end.
The action was fast and lively but there was a lack of penetration from both sides in the first half.
Turnbull saved well from Stephen Ireland early in the second half before the penalty drama in the 53rd minute.
Tuncay was not far off target with a fierce shot from outside the penalty area.
City piled on the pressure late in the game as Boro approached the nervous 90s. No Premier League team has conceded more late goals than Middlesbrough this season, so the relief of their fans was obvious when Gary O'Neil crashed in a stoppage- time goal to clinch the points.
Turnbull turned in another impressive performance to justify the faith of his manager in him.
Southgate refused to go into the transfer market in the summer for a new goalkeeper and Turnbull, a product of the club's highly successful academy, has come up trumps.