The visitors lost wickets at regular intervals in their run chase but were always up with the required rate and, with the scores tied and one ball to go, Babar launched Samuels over the mid-off boundary to seal the win.
Kieron Pollard (49 not out) showed a welcome return to form as he anchored the West Indies innings after they had won the toss and elected to bat.
Pakistan lost early wickets in their reply, but Umar Amin, also playing his first T20 international, smashed 47 in 34 balls before he was stumped off the bowling of Samuels.
Shahid Afridi took Pakistan close, scoring 46 from just 27 deliveries, including launching his 400th six in all international cricket.
With Pakistan needing six off the last over, Samuels took his team to the brink of a super over, before bowling a full toss that Babar smashed for six from the final delivery.
"It was a great effort from the players, I was really happy to see the desire to win and really happy with the contributions of the debutants and Shahid Afridi. It was a tough chase, we had to be positive," Pakistan captain Mohammad Hafeez told a television interviewer.
The second and final match of the series will be played at the same venue on Sunday.
(Writing By Nick Said in Pretoria; Editing by John Mehaffey)
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