Kohli hit an unbeaten 114 and Iyer 65 as they made a challenging chase of 255 from 35 overs look easy. Kohli hit the winning runs from his 14th boundary for India to reach 256-4 with 15 balls to spare.
The West Indies posted 240-7 around a three-hour rain delay which cut 15 overs from its innings. Before the rain, Chris Gayle smashed a crowd-pleasing 71 from 41 balls in possibly his last ODI, and Evin Lewis added 43 from 29. Their opening stand of 115 came at a run rate of nearly 11.
But they were out to catches in consecutive overs, and only Nicholas Pooran, with 30 from 16, lashed out like they did. Medium-pacer Khaleel Ahmed took an expensive three wickets.
Just like in the second ODI on Sunday on the same ground, Iyer joined his captain with India wobbling at three wickets down. And just like on Sunday when they combined for 120 runs, they hit a match-deciding 125 together.
Iyer relieved India nerves by swatting 65 from 41 balls, including five sixes. After his second straight fifty, he was caught at long off with India on 211-4 and cruising.
Three days after knocking 120, Kohli played the anchor and brought up his 43rd ODI century, only six from Sachin Tendulkar's world record.
Kohli finished on 114 not out from 99 balls, receiving man of the match again and man of the series. He was dropped early behind the wickets by Shai Hope, and batted in pain after a lifter from Kemar Roach hit his hand. Afterward, Kohli said he suffered a split nail and didn't believe his thumb was broken.
Kohli joined his teammates in giving Gayle an informal guard of honor after catching out the cricket great with a dive at mid-off.
Gayle wore jersey number 301 to mark his 301st and possibly last ODI, and entertained a sparse crowd by toying with the West Indies attack for 12 overs. He hit five sixes, including one one-handed over long leg off Ahmed.
He and Lewis shared 114 in the first 10 overs, the highest 10-over score by a team batting first in ODI history. If this was his last ODI, Gayle finished in style.
"It didn't work out for us today, we batted well up top and credit to Chris and Evin for the start they gave us," West Indies captain Jason Holder said. "They were positive from the start and showed what they are capable of.
"I was really pleased how we batted the situation after the rain break. Before, on previous occasions, we might have gone helter-skelter and lost wickets. Today, we handled the situation really well and we put ourselves in a position to win."
Gayle hasn't been picked for the two-test series, which starts on Thursday next week in North Sound, Antigua.