West Indies captain Pollard became only the third man to strike six maximums in an international.
In his previous over, Dananjaya had removed Evin Lewis, Chris Gayle and Nicholas Pooran.
West Indies chased their target of 132 in 13.1 overs in Antigua.
The tourists had been restricted to 131-9 before the drama really began.
In the fourth over, with the Windies already having 52 on the board, off-spinner Dananjaya had Lewis caught at long-off, Gayle - in his first international for two years - lbw and Pooran caught behind.
In Dananjaya's next over, Pollard climbed into the bowling, clearing the ropes in the arc between long-off and deep mid-wicket.
Watch Pollard's carnage here:
Absolute scenes 🤯@KieronPollard55 becomes the first @windiescricket player to hit six straight sixes in a T20I!#WIvSL pic.twitter.com/nrtmJHGcip
— ICC (@ICC) March 4, 2021
Pollard repeated the feat achieved by South Africa's Herschelle Gibbs against the Netherlands at the 2007 50-over World Cup, and India's Yuvraj Singh against India at the Twenty20 World Cup in the same year.
Pollard's fellow West Indian, the great Sir Garfield Sobers, was the first man to hit six sixes in a professional game, doing so for Nottinghamshire against Glamorgan in Swansea in 1968.
"I felt I could hit six sixes after the third one," said 33-year-old Pollard, who will play for Welsh Fire in this summer's inaugural Hundred competition.
"Once I had five sixes I knew I had the bowler on the back foot. He was going round the wicket and it was difficult for him. I just told myself: 'Go for it'."
Yuvraj Singh took to Twitter to welcome Kieron Pollard to an elite group of players who have hit six sixes in an over.
Welcome to the club @KieronPollard55 #sixsixes you beauty !!!⭐️ 🌟 ⭐️🌟⭐️🌟
— Yuvraj Singh (@YUVSTRONG12) March 4, 2021
Gibbs also went on to point out the Mumbai Indians connection as he took to Twitter to congratulate and welcome Kieron Pollard. Notably, Gibbs, Yuvraj and Pollard have been part of the Mumbai Indians franchise in different seasons of the Indian Premier League.
"1.2.3 @mumbaiindians representing first 3 to hit 6x6s in international cricket. Welcome @kieron.pollard55 Fisted hand #6x6 #goinggoinggone.#yuvrajsingh #kieronpollard #mumbaiindians," Gibbs tweeted.
1.2.3 @mumbaiindians representing first 3 to hit 6x6s in international cricket. Welcome @kieron.pollard55 👊 #6x6 #goinggoinggone ✅✅✅✅✅✅ #yuvrajsingh #kieronpollard #mumbaiindians https://t.co/FVBr6Rq60q
— Herschelle Gibbs (@hershybru) March 4, 2021
Sri Lanka skipper Angelo Mathews hailed his opposite number.
"Pollard as we know is very destructive," Mathews told the post-match presentation.
"What I told Akila was to go for his wicket even if he gets hit for six sixes. If he miscued one we could have got him out. Unfortunately it didn't happen."
Only eight men have hit six sixes in an over in all forms of cricket with fellow West Indian Garfield Sobers the first to do so in an English county game in 1968.
