The left-handed Axar (34) lifted Delhi from a precarious 62-5 to 144-9 in a gritty partnership of 69 with Manish Pandey (34) on Hyderabad's home turf.
Axar then returned figures of 2-21 with his left-arm spin.
Fast bowler Mukesh, who came in under the new impact player rule, defended 12 off the final over to keep Hyderabad down to 137-6.
It was Delhi's second successive victory after starting the season with five losses. The David Warner-led team stayed bottom of the 10-team heap.
"I love it here. It's an amazing crowd. I haven't been here for a long time," Warner, after playing at his former franchise, which he led to victory in 2016, said.
The Australian opener said, "To win tonight, hopefully, we can make it three in a row. We've got another against Sunrisers next."
Hyderabad, led by Aiden Markram, are one place above Delhi on ninth with four points and five losses.
Delhi's South African fast bowler Anrich Nortje dented the Hyderabad chase as he bowled Harry Brook bowled for seven. Axar soon joined in to cut short Mayank Agarwal's knock on 49.
Pace spearhead Ishant Sharma and left-arm wrist spinner Kuldeep Yadav picked up a wicket each to rattle the opposition middle order before Heinrich Klaasen and Washington Sundar attempted to revive the chase.
Klaasen fell to Nortje after a 19-ball 31 and Sundar, a left-hand batsman, made an unbeaten 24 but Mukesh and Delhi had the last laugh.
Delhi elected to bat first but lost half their side, including Warner who hit 21, in eight overs before Axar and Pandey turned the innings around.
"I ordered for coffee and I left the glass like that, when three wickets fell in one over," man of the match Axar said with a smile. "Pandey and I discussed we need to take it as deep as possible."
Sundar returned figures of 3-28 with his mystery spin as he struck three times in one over including the wicket of Warner.
Seam bowler Bhuvneshwar Kumar took two wickets and the innings witnessed three run-outs.
