Pogacar, who won the 2020 and 2021 Tours, attacked from the closing peloton on a steep hill near the finish line of the second stage as he, Jonas Vingegaard and Remco Evenepoel all gained time on the day.
"I was so close to taking the yellow jersey last year but then I cracked," said the 25-year-old Slovenian.
"Maybe this is confirmation I'm the strongest.
"Jonas was quite fast on my wheel, but I'm glad I blew that peloton open, took some time off some people."
Overnight leader Romain Bardet, winner of Saturday's opening stage, had predicted he would be unable to defend the yellow jersey and so it proved as he was dropped on the final ascent.
A 10-man attack had led as the race arrived in Bologna through apricot and peach orchards from the coastal resort of Cesanatico.
Vauquelin, making his Tour debut, broke away from the remnants of the escape on the final hill, and won solo at a packed finish line in the city centre with temperatures hitting 33C (91F).
"It was painful, but I did it for myself, my family and my team," said the 23-year-old winner who had lost 30 minutes on the opening stage.
"This is mad, yesterday I was on the bottom rung of the ladder. But today was perfect, how mad is that?"
Evenepoel at the party
Behind them Pogacar produced a blistering acceleration on the very steepest part of the final climb, but was immediately shadowed by defending champion Vingegaard of Denmark.
This pair were then joined downhill by former Vuelta winner Evenepoel and former Giro winner Richard Carapaz.
"I made a mistake, I shouldn't have taken Carapaz with me," said Evenepoel.
While Carapaz led that group across the line, official results gave Pogacar the yellow jersey, with the others on the same time.
"I managed to close the gap, but I could have done better," said Evenepoel, who took the best young rider's white jersey.
While the Quick Step man did himself down, he received high praise from the man in yellow.
"He's some rider, it was no surprise to see him get back on my wheel," said Pogacar.
Along the road Sunday were many skull and crossbones flags in memory of former Italian rider Marco Pantani, whose 1998 Giro d'Italia and Tour de France double Pogacar is hoping to match this year.
Pantani, who faced a barrage of doping allegations during his career and later died aged 34 from a cocaine overdose, was born in the stage's start town. His parents were guests of honour for the day.
Sprinters get reprieve
After two stages featuring some punishing climbs, stage three is a lengthy but flat 231km run from Piacenza to Turin that will have the sprint teams on red alert all day.
Jasper Philipsen, the defending green jersey wearer for the best sprinter, said ahead of the Tour there were nine interesting stages for sprinters, starting with Monday's.
Two days after "seeing stars" and vomiting with heat sickness 39-year-old Mark Cavendish will continue his bid for a record 35th stage win.
And Mads Pedersen, Sam Bennett, Wout van Aert and Fabio Jakobsen should also feature in any eventual bunch sprint in the industrial city home to motor giant Fiat and football legends Juventus.
