Alves has penned a four-and-a-half year contract at The Riverside after Boro agreed to pay a club record fee of just over £12million.
At one point Alves looked as if he might be forced to join AZ Alkmaar, who he signed a deal with last month.
But the Dutch Football Association refused to sanction the move which left Heerenveen free to agree a deal with Boro - which was finally done on Thursday evening.
Speaking to his official website, Alves admitted his delight at sealing his move to Teesside.
"I'm very happy to be able to play in such an important league, the biggest in Europe," said Alves.
"I am hoping that I can have a good performance, and repeat what I achieved in Sweden and Holland.
Boro manager Gareth Southgate told his club's official website: "I'm delighted to have completed the deal.
"As usual at this time of the year, it was a late night - but Afonso brings us something different in that he is a goalscorer who does much of his best work inside the box."
Alves was in the crowd at the Riverside for Tuesday's 1-0 Barclays Premier League victory over Wigan, having started the process of undergoing his medical.
His scoring record for Heerenveen was phenomenal, smashing in 45 goals in 38 appearances - including seven in one game earlier this season.
He joined Heerenveen in the summer of 2006 from Swedish side Malmo.
"His goalscoring record - in Holland, in particular, but in Sweden too - has been prolific," Southgate added.
"We hope he will reproduce that sort of form for us in the Premier League."