Kaneria dismissed a stubborn Shivnarine Chanderpaul (69) and Daren Powell next ball in the second over after tea to help send the tourists on their way for 244 in their second innings.
Pakistan won the first Test by nine wickets in Lahore. The second Test ended in a draw at Multan.
"We played well throughout the series and the result is ample proof of it," said Pakistan captain Inzamam-ul-Haq, under whom Pakistan also beat India and Sri Lanka this year.
West Indian captain Brian Lara said the defeat was "disappointing" after the visitors came back strongly at Multan to salvage a draw.
"I am still proud of my players," said Lara, whose team has not won a Test abroad in 45 attempts.
"Things have not gone well for us in Test cricket but we can take a lot of positives from this series. Pakistan were the deserved winners of the series," added the West Indian captain.
The 32-year-old Chanderpaul held the innings together after the West Indies lost the key wicket of Lara (49) and then vice captain Ramnaresh Sarwan (35) had to retire hurt with a foot injury.
The Guyanese hit ten boundaries during his fighting 155-minute knock and shared valuable partnerships with Dwayne Bravo (26) and Denesh Ramdin (25 not out) before Kaneria trapped him in front of the wicket.
Kaneria finished with 3-69, giving him 14 wickets in the series.
The home team took the last four wickets for 18 runs in the space of 68 deliveries to achieve victory.
Set an improbable 444-run target, the West Indies survived the first 90 minutes of the extended two-and-a-half hour session as skipper Lara and Sarwan batted cautiously after resuming the day at 39-2.
Lara, who needed 96 to become the first man to score 12,000 Test runs, was dismissed off an uppish drive off Umar Gul, which was smartly snapped by Shoaib Malik at mid-off. Lara hit nine boundaries during his 92-ball knock.
The Lara-Sarwan stand for the third wicket yielded 80 runs.
Sarwan, who regained his place in the team after being dropped in the second Test, had to return to the pavilion after he was hit on the right foot by a full-length delivery from Gul.
Sarwan, who went to hospital for an X-ray, had made 35. His report revealed a hairline fracture and he was ruled out of the one-day series which follows the Test series.
Morton hit two boundaries and a six off Danish Kaneria before giving the Pakistani leg-spinner a return catch to leave the West Indies at 144-4 at lunch.
After the break, Nazir dismissed Bravo with an edge to the slips to break the stubborn fifth-wicket stand and raise hopes of a Pakistan win.
After Kaneria's double strike and Abdul Razzaq's removal of Jerome Taylor for one, Pakistan were left with just one wicket to take.
Razzaq then trapped Corey Collymore leg before for nought to wrap up the innings with Sarwan unable to resume. Razzaq took 2-23.
The West Indies' target was always tough as no team in history has chased more than the 418 which the West Indies achieved for the loss of seven wickets against Australia in St John's in 2003.
But it was the big-scoring Mohammad Yousuf, who deservedly won the man-of-the-match and man-of-the-series awards, who was the difference between the two teams with his 665 runs in three Tests.
Yousuf hit a century in each innings in the third Test and broke the 30-year-old world record of West Indian great Viv Richards with 1,788 Test runs in a calendar year. He also set a new record with his nine centuries in 2006.
Pakistan and the West Indies start the five-match one-day series with the first match in Rawalpindi on Dec 5.