Klaasen put on key stands, including an 88-run partnership with Ishan Kishan, who made 55, to guide Hyderabad to 235-4 at their home Rajiv Gandhi International Stadium.
Klaasen and Kishan built on the blazing start by openers Abhishek Sharma and Travis Head, the pair putting on 54 runs off 21 balls.
Abhishek struck 35 off 13 balls, and Head hammered a quick 38 to lay the foundation for the total.
Hyderabad's bowlers restricted Punjab to 202-7 despite a valiant unbeaten 107 by Australia's Cooper Connolly, who struck his maiden T20 century in his debut IPL season.
Connolly, 22, surpassed Quinton de Kock's record as the youngest overseas player to score an IPL hundred. South African De Kock was 23 when he made a century for Delhi a decade ago.
Hyderabad skipper Pat Cummins was full of praise for his compatriot.
"Good to see the Aussies doing well. He's been a star this season with the bat for them. Yeah, not bowling at the moment, but still showing his worth as a batter. So good time for Oz," said Cummins, who took 2-34 and a pair of catches.
Hyderabad knocked Punjab from top spot and took over first place in the table with 14 points from 11 matches.
Punjab, who are eyeing their first title, slipped to second with their third straight defeat, but have played one match fewer than 2016 champions Hyderabad.
"We weren't that comprehensive enough on the field, nor in bowling, nor in batting," said Punjab skipper Shreyas Iyer.
"So I think they played comprehensive cricket, and they showed us how to win the match. We got to go back, reflect on things that we didn't do well in the last three matches, and see to it that we come back strong."
Sharp fielding gave Hyderabad the advantage, with secure catching contrasting Punjab's missed chances.
Connolly dropped Kishan on nine, and the wicketkeeper-batter survived again on 18 when Lockie Ferguson spilt a chance at backward square-leg.
Kishan lived a charmed life as Prabhsimran Singh missed an easy stumping off Yuzvendra Chahal. Shahshank Singh dropped Klaasen on nine.
The missed opportunities allowed both batters to pass fifty. Klaasen finished strongly in his 43-ball knock before being dismissed on the final ball of the innings.
Punjab were never in the chase after Hyderabad skipper and pace bowler Pat Cummins dismissed Priyansh Arya for one in the first over, and Nitish Kumar Reddy removed Prabhsimran in the next.
Wickets kept tumbling as Marcus Stoinis (28) and Suryansh Shedge (25) had their knocks cut short by Shivang Kumar and Cummins.
Connolly waged a lone battle and put on 68 runs with Marco Jansen (19), but the partnership could only reduce the margin of defeat for Punjab.