Stoinis and Kyle Mayers each scored half-centuries (54) on Friday, while Ayush Badoni and Nicholas Pooran added to the total with 43 and 45 as Lucknow ended on 257-5, six runs short of the record of 263 set by Royal Challengers Bangalore in 2013.
Mayers got Lucknow off to a strong start, combining with captain KL Rahul for 41 runs, before Stoinis and Badoni put together an 89-run stand for the third wicket.
Stoinis has batted at either number five or six for Lucknow this season, but the Australian excelled in an unfamiliar role on Friday, smashing 11 boundaries.
"I do enjoy the responsibility of batting a bit lower and seeing the game home," Stoinis said.
"Part of my journey is batting at every position. Bowl the first over, bowl the last over. I'm happy to do whatever my team needs."
The 33-year-old also contributed with the ball, dismissing opener Shikhar Dhawan in the first over of Punjab's innings, before his evening was cut short when he injured his finger, forcing him to retire hurt.
Asked about the finger, Stoinis said: "It's been better but yeah, it is what it is. We will get a scan later."
Lucknow pacers Yash Thakur and Naveen-ul-Haq picked up seven wickets to bring Punjab's chase to an end on 201 runs, with Atharva Taide top-scoring for the hosts with 66.
Lucknow next face Bangalore on May 1, while Punjab take on Chennai Super Kings on Sunday.
