Fraser-McGurk, who had celebrated his 22nd birthday on Thursday, shared a 77-run partnership for the third wicket with captain Rishabh Pant (41) to help Delhi overhaul Lucknow's 167-7 with 11 balls to spare.
Delhi moved to ninth in the 10-team contest after registering only their second victory from six outings.
Lucknow, who came into the match with three wins out of four, are now placed fourth.
"Spent five-six games on the sidelines itching to get out there," said Fraser-McGurk.
"I'm so happy to be here, different world in terms of cricket. Never seen anything like it, to be able to savour eight weeks, hopefully more, is amazing."
Delhi lost opener David Warner early with the Australian dragging a ball from Yash Thakur on to his stumps soon after he was attended to by the physio for a blow he took on his knuckles.
Prithvi Shaw (32) was caught brilliantly by Nicholas Pooran at deep midwicket off Ravi Bishnoi but Pant and Fraser-McGurk steadied the innings with their stand.
Fraser-McGurk, who went unsold in the IPL auction but came in as a replacement for Lungi Ngidi, hit Krunal Pandya for three huge sixes in a row in one over to underline his power-hitting prowess.
The Australian's 55-run knock, studded with five sixes and two fours, came to an end when he holed out to Arshad Khan, but by then Delhi needed just 28 runs off 32 balls.
Pant went past 3,000 IPL runs before being stumped off Bishnoi for 41 off 24 balls including two sixes and four boundaries.
"We were about 15-20 runs short, we should've capitalised to get to 180," said Lucknow skipper KL Rahul.
"McGurk was very unknown. We've watched a lot of videos of him but he hit the ball really well, credit to him."
Earlier, Lucknow were off to a fine start after electing to bat but they stumbled after the powerplay, losing two wickets in two balls to left-arm wrist spinner Kuldeep Yadav (3-20).
Yadav had missed Delhi's last two matches due to an injury.
The slow bowler struck to send back Rahul (39 off 22 balls).
Rahul tried to slash the ball which was turning away and a Delhi review confirmed a nick.
Lucknow were tottering at 94-7 in the 13th over but Ayush Badoni top-scored with an unbeaten 35-ball 55 and shared 73 runs for the eighth wicket with Arshad Khan (20 not out) to ensure a fighting total on the board.
It was the first half-century this IPL season for Badoni, who hit one six and five fours.
