Last year's runners-up Rajasthan posted 199-4 after an attacking 98-run opening stand between the left-handed Jaiswal (60) and Buttler (79) in their second home at Guwahati.
New Zealand's left-arm quick Trent Boult led the bowling charge with a double-wicket maiden over to restrict Delhi to 142-9 as Rajasthan bounced back from their previous loss.
Boult and leg-spinner Yuzvendra Chahal took three wickets each.
Rajasthan, who won the first edition of the Indian Premier League in 2008 under late Australian great Shane Warne, have won two of their three matches to lead the 10-team T20 tournament.
David Warner's Delhi slumped to their third loss of the season in as many matches.
The left-hander made 65 and got past 6,000 IPL runs but his valiant knock got little support from the other end.
Delhi's Rovman Powell fell for two after he made the XI in place of Australian import Mitchell Marsh, who flew back home for his wedding and will return in a week.
Jaiswal provided Rajasthan with an explosive start after being invited to bat first, the rising Indian opener smashing five boundaries in the opening over.
Buttler, who batted with stitches on his finger after getting hurt while fielding in the previous match, soon took over as he hit Anrich Nortje for three straight fours in the next over.
Rajasthan lost their way in the middle but Buttler, who reached his second 50 of the season with a six, soon changed gears with Shimron Hetmyer (39 not out).
Buttler, who got the orange cap for leading the season's batting chart, was later substituted by spinner Murugan Ashwin under the new impact player rule.
