The 19-year-old Spaniard, the youngest ever player to top the rankings, is looking to complete the "Sunshine Double" after beating Daniil Medvedev in last week's Indian Wells final.
Alcaraz made a blistering start winning the first 11 points of the match, with the 33-year-old Bagnis having no answer at all to the teenager's powerful serve.
There was a loud cheer from the crowd at Hard Rock Stadium when Bagnis, who has never beaten a top 20 ranked player, held serve in the second set and then broke Alcaraz to make it 2-2.
But that was a brief moment of respite for Bagnis rather than any shift in momentum and Alcaraz had no further problems as he blasted his way to victory in just 65 minutes.
"To start a new tournament is never easy, different conditions. I have to be ready to get used to these new conditions, but I was really focused on the match from the beginning, and I’m happy with the way that I played," said the Spaniard.
Alcaraz needs to win here to maintain his top spot in the rankings, will face either Serbian Dusan Lajovic or American Maxime Cressy in the third round.
After his win in Miami last year, Alcaraz boldly predicted he was ready to win a Grand Slam and he returns now having achieved that goal and, he concedes, as a different man.
"I grew up a lot since last year. It has been a great year for me as a player, as a person as well. I learned a lot of things, not only on court, also off the court, as well," he said.
His whirlwind year caught up with him physically - he had to miss January's Australian Open with a leg injury - and the Spaniard says he is now more careful about his physical condition.
"I have to take care about myself a little bit more off the court, taking care about going early to bed, to rest better, eat well, take care of myself off the court. That's the most important thing for me," he said.
- Fritz advances -
It was also a straightforward start to the tournament for America's top ranked player, world number 10 Taylor Fritz who saw off 21-year-old Californian Emilio Nava 6-4, 6-1.
Nava's ultra-attacking style had earned him an upset win over John Isner in the previous round and he hit some spectacular winners early against Fritz.
But Fritz kept his calm and with the first set under his belt, picked off Nava with ease in the second.
Fritz felt Nava, a promising talent, had made a common mistake of young arrivals on the tour.
"I think that he should just play his game more. In these type of matches I feel like he maybe felt like he had to play much higher than his capacity in this match," he said.
"Just the way he was just kind of going for winners off everything in the first set. It was working, but it's tough to sustain. That's something that I did as well when I was a kid," he added.
Germany's Alexander Zverev's postive return to the tour after his ankle ligament injury, which saw him reach the last 16 at Indian Wells, was cut short by a straight sets 6-0, 6-4, loss to Japanese wildcard Taro Daniel.
The 13th seed Zverev, who has type one diabetes, had to take an insulin shot during the match as New York-born Daniel picked up another impressive win after his victory over Matteo Berrettini at Indian Wells and win over world number four Casper Ruud at the Mexican Open.
Ruud defeated Belarus's Ilya Ivashka 6-2, 6-3, while Italy's Jannik Sinner also progressed to the third round with a 6-4, 6-2 victory against Laslo Djere.
After reaching the final in his Miami debut in 2021 and reaching the quarter-finals last year, the 21-year-old Italian will be looking to go deep again.
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