Opening batsman Prabhsimran's 103 off 65 balls guided Punjab to 167-7 after being put in to bat first at Delhi's Feroz Shah Kotla ground.
Brar then returned figures of 4-30 with his left-arm spin as Delhi suffered a collapse to end on 136-8 despite a strong start from skipper David Warner, who hit 54 off 27 balls.
Punjab moved to sixth to improve their chances of making the final four for a spot in the play-offs. Delhi remain bottom of the heap and are out of the running.
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Warner seemed to take the game away with his power hitting but Brar led the spin charge, removing Phil Salt for 21 as Delhi slipped from 69-0 to 88-6.
Leg-spinner Rahul Chahar joined in to trap Mitchell Marsh lbw before Brar struck twice in an over to dismiss Rilee Rossouw and Warner, who registered his 60th IPL fifty.
Impact player Nathan Ellis, who came in as a substitute for Prabhsimran, took two wickets with his pace bowling.
Earlier, Prabhsimran set the victory up for Punjab, smacking 10 fours and six sixes to play a lone hand in his team's total.
Punjab lost three early wickets including skipper Shikhar Dhawan, for seven, and England's Liam Livingstone, for four - both falling to senior pace bowler Ishant Sharma.
Prabhsimran stood firm and put on a key partnership of 72 with Punjab's expensive England import Sam Curran, who made 20.
The rest of the batting disappointed with Prabhsimran, who survived a dropped catch by Rossouw on 68, standing tall until his dismissal in the 19th over.
