Lucknow, chasing 156 for victory after Rinku Singh's unbeaten 83 in the middle order guided Kolkata to a competitive total, stumbled until they needed seven runs off the last ball of the innings.
With Mohammed Shami on strike, the Indian fast bowler smashed Kartik Tyagi's final delivery over long off to tie the match and send it into a Super Over.
However, the Super Over proved to be a masterclass from Kolkata's Sunil Narine, the 37-year-old dismantling Lucknow's batting in devastating fashion while conceding one run.
The experienced all-rounder dismissed fellow West Indian Nicholas Pooran off the first ball when he was bowled before Aiden Markram was caught in the deep off the third ball as Lucknow posted 1-2 in three balls.
"The entire team backed me (to bowl the Super Over), so I think once everyone said the same thing, it was pretty easy," Narine said.
"Obviously, bowling first, you could relax a little more, even though it's not a very relaxing place to be!"
With Lucknow in tatters, Rinku came in to bat again for Kolkata and wasted no time in settling the contest, smashing a boundary off the first ball to give Kolkata the win.
The result moved Kolkata up to eighth in the standings, while Lucknow are bottom with two wins from eight games.
"I think we definitely need a break. I think we're going to refresh. There is always pressure," Lucknow skipper Rishabh Pant said after they lost their fifth straight game.
"But, at the same time, we have to look (for) answers inside, not outside, and just keep it simple. Just take accountability, each and every guy... A lot of people will take accountability for that, for sure."