Brathwaite was again unbeaten on 60 as the West Indies passed their 153-run target without further loss
Brathwaite, who carried his bat in the first innings with 142 not out, was again unbeaten on 60 as the West Indies passed their 153-run target without further loss after beginning the fifth day on 114 for 5.
Shane Dowrich, who also finished with 60 not out, edged Mohammad Amir towards third man for a boundary as West Indies reached 154-5.
Pakistan won the three-match series 2-1 after winning the first Test in Dubai by 56 runs and the second by 133 runs in Abu Dhabi.
Brathwaite and Dowrich needed just 7.5 overs to score the 39 runs required.
Opener Brathwaite signalled his intent by driving fast bowler Wahab Riaz's first ball of the day through the covers for four.
Dowrich soon got in on the act when he pulled leg-spinner Yasir Shah to the boundary to reduce the target to 29.
Brathwaite pushed Riaz for two to reach his 12th Test fifty, finishing with six boundaries during his second unbeaten knock of the match, during which he faced 109 balls.
Dowrich drove Riaz through the covers for his fifth boundary to complete his fourth Test half-century. Two boundaries off Amir saw his side home.
"We showed character and fight," said West Indies skipper Jason Holder. "This has been a long time coming, We are a young side in transition. We've done a lot of things good this series."
Holder praised man-of-the-match Brathwaite.
"Credit must go to Kraigg. He played an outstanding innings in the first innings and took responsibility in the second and helped us cross the line," said Holder.
The result meant Misbah-ul-Haq could not celebrate his record 49th Test as Pakistan skipper with a win. He surpassed the previous mark of 48 in charge set by the legendary all-rounder Imran Khan.
"It's always disappointing to lose a Test," said Misbah.
"Credit to West Indies. They kept improving and we kept losing momentum. We were well below our standards, especially in the batting department."
It was Holder's first victory in 12 Tests in charge and a personal milestone as he collected his first Test five-wicket haul with 5 for 30 on Wednesday.
It was also the West Indies first win in 14 Tests since beating England at Barbados on May 3, 2015.
The 2-1 series win keeps intact Pakistan's unbeaten record in their adopted home of the United Arab Emirates since the 2009 attacks on the Sri Lankan team bus in Lahore forced them to play abroad.
Pakistan have won five and drawn four of the Test series they have played in the UAE.
West Indies' win did however prevent Pakistan from achieving a rare 9-0 clean sweep after winning both the Twenty20 and one-day International series 3-0.