The 17-year-old Parag from Assam played without any fear in only his 14th T20 match to score a 31-ball 47 in a determined rearguard innings as the Royals chased down the target of 176 with four balls to spare.
Parag, who hit five fours and two sixes, added 44 crucial runs with Archer -- who remained not out on 27 off just 12 balls with the help of two fours and two sixes -- for the seventh wicket from 3.3 overs to turn the match on its head after the Royals were reduced to 98 for 5 in the 13th over.
The Royals needed 54 runs from the final five overs with just four lower order batsmen to come and the match was definitely for KKR's taking at that stage but the script changed altogether as the home side slumped to their sixth loss on the trot.
Archer, who was playing his last IPL match of the season as he is leaving for England, hit a six to win the match as the Royals reached 177 for 7 in 19.2 overs.
The Royals made a flying start with openers Sanju Samson (22) and Ajinkya Rahane (34) scoring 53 off the first five overs but a sudden batting collapse saw then being reduced to 63 for 3 in the eighth over, losing three wickets in two overs for just 10 runs.
Spinners Sunil Narine and Piyush Chawla threaten to change the complexion of the game with a burst of two wicket apiece.
Captain Steve Smith (2) was out in the eighth over with Narine getting him.The Royals were looking down the barrel with the match drifting towards KKR before Parag and Archer enacted a memorable run chase and a win.
Earlier, under-fire skipper Dinesh Karthik silenced his critics with a career-best 97 not out as Kolkata Knight Riders recovered from an atrocious start to post a competitive 175 for six against Rajasthan Royals in their must-win IPL fixture.
Karthik seized the momentum in the 11th over when he hit Shreyas Gopal for a six followed by three boundaries in an over that yielded 25 runs and KKR did not look back since then.
Unadkat conceded 50 runs for his one wicket and was the most expensive of the lot as he undid all the good work done by Varun Aaron (2/20) who had a fine new ball spell of 3-1-10-2.
Put in to bat, KKR got off to an atrocious start to be 32/2 in powerplay overs against an all out pace attack from Rajasthan Royals.
Playing his only second match this season, Aaron extracted pace and swing from the fast-paced Eden track and removed both the openers Chris Lynn (0) and Shubman Gill (14) cheaply.