West Indies big-hitter Kyle Mayers top-scored with 51 and Stoinis contributed 21 in Lucknow's total on a tough batting pitch in Jaipur.
Stoinis then took the two key wickets of Yashasvi Jaiswal (44) and Jos Buttler (40). Fellow quick Avesh returned figures of 3-25.
"It was nice to get involved with the ball," said man of the match Stoinis.
"There were not exact plans to get them (Jaiswal and Buttler) out, it was about using the conditions and using my smarts as a bowler."
With 19 needed off the last over, Avesh kept his calm to strike twice and restrict Rajasthan to 144-6.
Rajasthan keep their top spot in the table with eight points, ahead of second-placed Lucknow, who also have four wins from six matches but stay behind on an inferior run rate.
Stoinis broke a promising opening stand of 87 between the left-handed Jaiswal and Buttler and denied both batsmen their fifties.
Last year's runners-up Rajasthan looked like coasting along before they lost Jaiswal, skipper Sanju Samson, run out on two, and Buttler in the space of three overs.
"That Samson run out and two wickets falling back-to-back lets you back into the game," said Lucknow skipper KL Rahul.
"Their top three-four are really strong and our plan was to get them out early, and it came via a run-out today."
Avesh dismissed the big-hitting Shimron Hetmyer but impact player Devdutt Padikkal took charge and attempted to revive the chase with his quickfire 26 but it was in vain.
Avesh got Padikkal caught behind in the final over and sent back Dhruv Jurel for nought to be on a hat-trick denied by Ravichandran Ashwin.
Rajasthan, inaugural IPL winners in 2008, began the match with a maiden over from New Zealand left-arm quick Trent Boult, who returned figures of 1-16.
Ashwin took two wickets, including the dangerous Mayers after the left-handed opener struck his third fifty of the season following a sluggish start.
Rahul was the first to go after an unconvincing 39 in an 82-run opening stand with the left-handed Mayers.
Stoinis and West Indies star Nicholas Pooran, who hit 29, put on 45 runs for the fifth wicket to help the team cross 150.
