Gayle hit four fours and two sixes in his 28-ball 40 as the Kings scored 221 for 6 in their 20 overs.
Gayle and KL Rahul put on 67 for the second wicket after they had lost Mayank Argawal for 14 with the score on 22.
Rahul and Deepak Hooda put on 105 for the third wicket, 64 of them coming from Hooda’s bat plundered four fours and sixes in his 28-ball bat.
Nicholas Pooran failed to trouble the scorers out first ball. Rahul eventually fell for 91 in the final over having hit seven fours and five sixes in his 50-ball innings. However, after has dismissed only one more run was scored from the remaining four deliveries.
Left-arm medium-pacer Chetan Sakariya claimed the wickets of Rahul Agarwal and Jyhe Richardson to return figures of 3-31 from his four overs.
Chris Morris, who got Gayle, took 2-41.
Needing 222 for victory, the Royals chase was led by Sanju Samson who a fantastic hundred. With his team needing five runs to win from the last ball, Samson was caught in the deep, going for a six over sweeper cover.
Arriving at the crease after Ben Stokes was out for a duck, Samson plundered the Kings bowling scoring 119 runs off just 63 balls during and hit 12 fours and seven magnificent sixes.
Unfortunately, he did not have much support as Jos Butler and Riyan Parag each had cameos before getting out for 25 but were unable to stay with Samson. Shivan Dube also got 23, but he, too, didn’t last long.
Asrshdeep Singh who got Samson’s vital wicket returned figures of 3-35. Mohammed Sami took 2-33.
The win means the home side’s seven-match losing streak continues as they ceded the three-match series to India with one match remaining.
India started slowly but eventually got the measure of the West Indies attack with half-centuries from Shreyas Ayer (63) and Sanju Samson (54) but it was Axar Patel whose quick-fire 64 from 35 balls that powered India over the line with a straight six from the fourth ball of the final over bowled by Kyle Mayers.
Mayers finished with 2-48 from 7.4 overs. Alzarri Joseph bowled well for his 2-46 from 10. Jayden Seales proved economical taking 1-40 from his allotment.
Earlier, after the West Indies had won the toss and chose to bat first, Hope scored 115 as the West Indies posted 311-6 from their 50 overs.
After sharing in an opening stand of 65 with Kyle Mayers who made 39, Hope also featured in a second-wicket partnership of 62 with Shamarh Brooks who contributed 32 before he too was dismissed after getting a start.
Brandon King made nought from five balls before Hope and Captain Nicolas Pooran shared in a 127-run partnership for the fourth wicket that got the West Indies back on track for another score over 300. Pooran was eventually dismissed for 74 with the West Indies positioned at 280-5 in the 44th over.
Hope finally fell in the 49th over, caught in the deep trying to hit over the long-off boundary as the West Indies' rate of scoring dipped. The home side managed 31 off the next six overs while also losing the wickets of Rovman Powell for 13 and Romario Shepherd for 14, something that West Indies would come to regret.
Shardul Thakur was the pick of the Indian bowlers with 3-54 from his seven overs but India’s ability to restrict the West Indies' scoring in the death overs was largely due to the excellent bowling of Mohammad Siraj, who ended wicket-less but who the West Indies batters found difficult to get away in the final three overs.
In a game that Sri Lanka started brightly, India ultimately proved too strong as they continued their impressive record over their opponents in this format.
India's openers endured a difficult start to the encounter, as Yashasvi Jaiswal (10) Sanju Samson (0) and Rinku Singh (1) all fell inside the first four overs.
Shubman Gill would top score for the Men in Blue with 39 runs from 37 deliveries as Maheesh Theekshana starred with the ball in hand, taking 3-28 as Sri Lanka restricted their opponents to 137-9.
Kusal Mendis (43) and Kusal Perera (46) would give Sri Lanka an excellent start to their chase, losing just two wickets in the first 16 overs of the contest.
But with time running out, India's bowlers stepped up the pressure and claimed six wickets from the final four overs. Needing just six runs from the final over, Yadav's bowling display forced a super over.
Washington Sundar was given the responsibility to deliver, and he claimed two wickets in three balls with Sri Lanka finished the super over having scored just two.
Yadav would waste no time in sealing the win, smashing a four from the opening ball to secure a 3-0 series win.
Data Debrief: India continue Sri Lanka dominance
India have now won five of their last six men’s T20Is against Sri Lanka (L1), including the last four on the bounce.
The Men in Blue have won all of their four men’s T20Is against Sri Lanka at the Pallekele International Cricket Stadium.
Sri Lanka have now lost their last three men's T20Is at Pallekele, the second time they have done so in their history (three loses from September 2016 to September 2019).
Sanju Samson struck 86 off 46 balls for the second-placed Royals, but the Capitals successfully defended their haul of 221.
Jake Fraser-McGurk's 20-ball 50 set the tone for the Capitals, who are well in the play-off hunt with two matches remaining.
Fellow opener Abishek Porel plundered 65, with Tristan Stubbs adding a useful 41 towards the end of the innings, with Ravichandran Ashwin (3-24) the only Royals bowler to do significant damage.
Kuldeep Yadav (2-25) and Mukesh Kumar (2-30) led the way with the ball for Delhi, with the latter taking the vital wicket of Samson in the 16th over, leaving the Royals with too much to do from 162-4.
Delhi have won three of their last four matches and are making a late run for the play-offs, as one of four teams tied on 12 points.
Data Debrief: Samson soars
It took Samson just 31 minutes to amass 86 runs before he was eventually dismissed, with the loss of his wicket all but ending the Royals' hopes.
In the process, he moved into third place in the IPL run-scorers chart, up from 11th, with 471 runs to his name this season. He trails only Ruturaj Gaikwad (541) and Virat Kohli (542).
The Royals have now won eight of their nine matches in the 2024 campaign and are almost certain to be involved in the playoffs after chasing down their victory target of 197 with an over to spare in a seven-wicket triumph.
Another Royals win had looked unlikely when they fell to 78-3 in their run chase after Jos Buttler (34), Yashasvi Jaiswal (24) and Riyan Parag (14) all fell in quick succession shortly before the halfway mark.
But Rajasthan captain Samson fought back as he blasted 71 runs from 33 balls including seven fours and four maximums, while Jurel bounced back from a poor period of form to finish unbeaten on 52 from 34.
LSG reached 196-5 from their innings courtesy of a fine knock from their own skipper, KL Rahul. He reached 76 from 48 before eventually falling to Avesh Khan (1-42) in the 18th over.
Deepak Hooda made it to 50 as LSG rallied from 11-2, but ultimately their competitive total was within reach of the visiting Royals.
Data Debrief
This is the second time in as many home games Rahul has passed fifty, after he scored 82 against Chennai Super Kings last week.
Samson (385) and Rahul (378) are second and third respectively in the race to win this season’s orange cap, although RCB’s Virat Kohli continues to set the pace with 430 runs.
They also narrowly trail Kohli (40) when it comes to the most fours hit in this season’s competition, with Samson on 36 and Rahul just behind with 34.
