After being set 211 to win when West Indies' second innings ended at 181, Pakistan slumped to the brink of defeat at 9-71 in the face of accurate bowling from Jayden Seales, who took 5-20 in Trinidad on Tuesday (Wednesday AEST).
Babar finally found a stoic partner in Abbas and the last pair held out for 12 overs to frustrate the hosts. Abbas was the last man out for 23 shortly after tea on the fourth day when Pakistan were 120.
Babar's half-century stood out in stark relief against a poor Pakistan batting effort. When he reached his half-century from 81 balls, the tourists were 9-88 and the rest of their batters had contributed only 38 runs.
"We played good cricket in the last few days, but today we were not up to the mark as a batting unit," Babar said.
"The surface is good and we expected the new ball to move for which we were prepared ... but I think they bowled very well and used the conditions nicely."
Pakistan's collapse followed the template of their first innings in which they reached 4-244, with a century by Shan Masood, then lost their next six wickets for 38 runs to be all out for 282, conceding a 29-run first innings lead.
Masood suffered a hand injury while batting in the first innings and was forced to drop from No.3 to No.8 in the order second time of asking.
West Indies resumed on day four at 7-126, with a lead of 155 runs. Shamar Joseph went from 22 not out overnight to 38 and Kemar Roach from five to 18 to push their lead beyond the psychological barrier of 200 runs.
That proved more than adequate as Pakistan lost Imam-ul-Haq (three) in the third over to set the visitors' second collapse of the match in motion.
Babar provided solidity at one end but, of Pakistan's top six batters, only Mohammad Rizwan with 11 reached double figures.
Pakistan folded in the face of Seales' early assault and contributions of 2-27 from Roach and 2-12 from Justin Greaves. Seales returned to make the final breakthrough, trapping Abbas lbw in the 41st over to finish with his fifth five-wicket bag in Tests.
"We came in and tried to be just as disciplined as possible," Greaves, who took 5-27 in the first innings and was named player of the match, said.
"I mimicked Mohammad Abbas' fuller lengths when he bowled against us in the first and second innings and it pretty much worked out for us."
Abbas took 3-63 and 5-22 for Pakistan.
