Needing a further 39 runs on the last day to deny Pakistan a clean sweep in the series, Brathwaite and Dowrich calmly steered the tourists to their victory of target of 154 with both batsmen finishing with identical scores of 60 not out.
The pair shared an unbroken sixth wicket partnership of 87 after coming together on Wednesday with the West Indies in trouble at 67 for five.
Pakistan won the opening test, played as a day-nighter, then the second by 133 runs but the third test defeat cost them a rare triple cleansweep after they won the Twenty20 and ODI series' 3-0.