Khan and Ahmad took five wickets between them as Rajasthan were bowled out for just 118 in the 18th over of their innings after Rajasthan captain Sanju Samson won the toss and decided to bat first.
Royals were rocked early and lost both openers - England's Jos Buttler (eight) and Yashasvi Jaiswal (14) - by the sixth over.
Samson, who hit a fluent 20-ball 30, was their top scorer and his dismissal in the seventh over left Rajasthan reeling at 60-3.
Khan took three wickets to remove Ravichandran Ashwin (two), Riyan Parag (four) and West Indian Shimron Hetmyer (seven) to finish with an impressive 3-14 in his four-over spell.
He was ably supported by Ahmad who dismissed Devdutt Padikkal (12) and Dhruv Jurel (nine), to finish with 2-25 in three overs.
New Zealand bowler Trent Boult (15) was Rajasthan's second-highest scorer before he was dismissed by Mohammed Shami.
India's Shami and Ireland's Josh Little took two wickets in eight tight overs between them.
Gujarat opener Shubman Gill (36) was removed by leg-spinner Yuzvendra Chahal in the 10th over before skipper Hardik Pandya hit an explosive 15-ball 39 not out with three sixes to wrap the game up in the 14th over.
Opener Wriddhiman Saha was also unbeaten on 41.
The win put reigning IPL champions Gujarat Titans three points ahead at the top of the table after 10 games.
Samson described it as a "tough night" and said that fourth-placed Rajasthan will "have to pull our socks up".
"With 120 to defend, that was a bit too much to ask for."
