Chase's 49 not out off 47 balls included two sixes and the winning boundary as West Indies scored 5-184 in 33.2 overs on Sunday (local time).
The hosts were chasing a revised target of 181 runs in 35 overs under the DLS method after multiple rain interruptions saw Pakistan's innings end at 7-171 in 37 overs.
Right-arm fast bowler Jayden Seales took 3-23 in seven overs.
West Indies stumbled from 3-101 after 18 overs at Brian Lara Stadium in Tarouba, Trinidad and Tobago, to 5-111 off 24 overs, including the wicket of Sherfane Rutherford, who hit a 33-ball 45 (three sixes, four fours).
Chase, named man of the match, and Greaves (26 off 31) shared a 77-run unbroken partnership for the sixth wicket.
West Indies earlier chose to bowl first and Hasan Nawaz again played a key role for Pakistan with 36 not out off 30 deliveries, after his unbeaten 63 in the ODI opener.
Hasan's knock on Sunday included three sixes - two of them in what turned out to be the last over of Pakistan's innings after more rain.
Hussain Talat made 31 off 32 balls, but most Pakistan batters were much slower, including captain Mohammad Rizwan, trapped lbw by Gudakesh Motie for 16 off 38 balls.
Pakistan beat West Indies in the first ODI on Friday, also by five wickets.
The third and final match in the series is scheduled for Tuesday at the same stadium.
Pakistan won the preceding Twenty20 series 2-1 played in Florida.
