The Australian No.1 was stretched by an hour-long opening set but once he shook off his nerves, got his feet moving and found his groove, he raced to victory on Tuesday.
Winning 12 games out of 13 after the first set tiebreak, the 27-year-old earned a 7-6 (7-5) 6-1 6-0 victory, wrapping up the final two sets in 52 minutes.
"A little bit of a nervy start," said the fifth seed. "Never easy starting an incredible tournament like Wimbledon, but as the match went on, I played better and better.
"I love being here. I really want to do well here. It's about getting started and I'm glad I got through that. I'm super-stoked to be in the next round.
"I love the grass. I love these courts. I am at my most dangerous when I am aggressive and it's up to me to be aggressive and try and dictate. Next time, hopefully I can do that from the start."
Coming on to court 3 after his fiancee, British No.2 Katie Boulter, had slumped to a shock defeat to an Italian qualifier, de Minaur seemed to inherit her disappointment.
Suffering an early break he quickly found himself 3-0 down against a resourceful opponent who belied the South American cliche of clay court-bred baselining.
It was 40 years and a day since Burruchaga's father, Jorge, scored the winning goal in the 1986 World Cup final, a 3-2 victory for Argentina over West Germany, and on his Wimbledon main draw debut the son seemed to draw inspiration from history.
De Minaur scrapped his way to a tie-break and, helped by a lucky net cord and Burruchaga double-faulting at 4-4, he managed to win it.
That released the nerves and de Minaur, seizing the initiative, suddenly ran away with the tie to book a second-round date with idiosyncratic French southpaw Adrian Mannarino.
Ranked 40, the 38-year-old is a dangerous opponent on grass but de Minaur has a 5-1 win record against him.
De Minaur was the first Aussie men's winner after an opening day when Aleksandar Vukic and Adam Walton went out.
Rinky Hijikata had been one set-all and 5-3 up in the third against Dutchman Jesper de Jong when dew and poor light forced a suspension on Monday night.
Hijikata was broken soon after Tuesday's resumption, but broke back to take the third 7-5. However, he lost the next two sets 6-4 6-3 to suffer a bleak loss.
There was heartbreak, too, for Thanasi Kokkinakis who twice led 10th seed Alexandre Bublik only to fade away, losing 4-6 6-3 7-6 (12-10) 6-3 6-4.
When the South Australian won a gripping 22-point third set tie-break it seemed a notable victory was in his grasp but in the final hour of a 3:39 duel the Kazakh proved too consistent.
Daria Kasatkina was Australia's lone first day victor but will be joined in the women's second round by Kim Birrell who defeated Russian qualifier Alina Korneeva 6-3 0-6 6-2.
However Talia Gibson went out. After a torrid opening set, she had fought back to level against 21st-seed Marie Bouzkova but the Czech's greater quality told in the third as the Perth player went down 6-1 3-6 6-2.
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