Kaur (51) produced a late onslaught on the Velocity bowlers after the Supernovas made a heavy weather of a small chase of 122. The Supernovas overhauled the target off the last delivery of the run chase, reaching 125 for 6 from 20 overs.
Coming out to bat in the 11th over at the team score of 53 for 2, the India T20 captain struck a quickfire 51 off 37 balls which was lashed with four boundaries and three sixes to take her side to the threshold of victory.
The Supernovas needed seven from the final over but drama unfolded as Kaur was out in the second delivery which raised the visions of a Velocity win. But Lea Tahuhu (2 not out) and Radha Yadav (10 not out) kept their nerves to hit the required runs in a tense finish. Radha Yadav hit a four off the last delivery to clinch the match.
The Supernovas struggled initially with the Velocity bowlers not allowing them score runs easily.
Opener Chamari Atapattu was run out for 2 in the second over itself while going a third run but half centurion Jemimah Rodrigues (22) and Priya Punia (29) steadied the run chase with a 44-run stand for the second wicket.
The runs were hard to come by with the Velocity bowlers stifling the Supernova batswomen. The Supernovas were dealt a blow as Rodrigues and Punia fell in consecutive deliveries to leave them in trouble.
Rodrigues fell to Amelia Kerr who took the catch off her own bowling off the final delivery of the 10th over while Punia was dismissed by Vaidya in the first ball of the 11th over.
The Supernovas were 53 for 2 at the halfway stage, needing 69 from 10 overs. Pressure began to pile on Supernova.
Bangladesh bowler Jahanara Alam then removed Natalie Sciver (2) and Sophie Devine (3) in her consecutive overs (14th and 16th) to reduce Supernovas to 64 for 5 in the 16th over.
The Supernovas needed 45 runs from the final five overs and Kaur showed why she is considered the premier batswoman in the country with a flurry of boundaries. She hit two fours off Jahanara Alam in the 16th over and followed it up with two sixes off Devika Vaidya in the 18th over to take her side near the victory target.
Earlier, Supernovas produced a clinical bowling display to restrict Velocity to 121 for 6.