The Northern Irishman, who had sought the six-time Masters winner out a fortnight before the Masters, threw them in but won anyway.
Nicklaus led the plaudits after McIlroy defied another Augusta National meltdown to beat Justin Rose in a playoff on Sunday.
His victory was the first in 17 Masters starts and completed the box set, McIlroy the first since Tiger Woods in 2000 and sixth in total to win all four of men's golf majors.
Tipping McIlroy after his ceremonial tee off on Thursday morning, Nicklaus added a warning that he had a history of making costly double bogeys.
McIlroy ended up with four across the tournament, more than any other victory before him in 89 Masters.
Nicklaus was among the first to publicly congratulate him after a mad final day that saw him give up a two-shot lead, create then relinquish a four-shot lead then yo-yo with Rose until a birdie clinched it on the first playoff hole.
"I am delighted for Rory. He has had the world on his shoulders and I am delighted to see him win." he said.
"Now we are going to see a lot more really good golf out of Rory McIlroy."
Woods was the last man to achieve the career grand slam 25 years ago.
"Welcome to the club @McIlroyRory," Woods posted on social media.
"Completing the grand slam at Augusta is something special. Your determination during this round, and this entire journey has shown through, and now you're a part of history. Proud of you!"
McIlroy's triumph was a "marvellous national occasion", Taoiseach Micheal Martin has said.
Paying tribute to the County Down golfer, the Irish premier said he had created a "moment in time that we will all remember".
He added: "For him, it's a very special moment and for the rest of the nation, we were privileged to share in it."
McIlroy's close friend Shane Lowry joked that McIlroy could happily retire after completing the career grand slam.
"This is all he thinks about, all he talks about," he said.
"He always said to me he'd retire a happy man if he won the green jacket so I told Erica (McIlroy's wife) he can retire now."
Tommy Fleetwood, who finished 21st after a closing 69, added: "I've said it a bunch of times, I think Rory is the best player of our generation. I don't think he's had to prove that for a while."
Victory was extra sweet for McIlroy and caddy Harry Diamond, a friend since they were seven-year-olds who has been on the bag since 2017.
Critics questioned the move time and time again, asking whether Diamond was the best man for the job.
But McIlroy stayed staunchly loyal to Diamond and refused to make a change, even though he hadn't won a major since 2014.
"We've had so many good times together," he said.
"He's been like a big brother to me the whole way through my life.
"So to be able to share this with him after all the close calls that we've had, all the crap that he's had to take from people that don't know anything about the game, yeah, this one is just as much his as it is mine."
With PA
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