The muscular Alcaraz, who skipped last year's tournament with a right leg issue, appeared overly exuberant at times as he sought quick progress at Melbourne Park but composed himself to grind down his seasoned opponent and blow him away.
"It's always great to play here in Australia, it's the third time that I'm playing here," said Alcaraz, who will meet Italian Lorenzo Sonego in the second round.
"I didn't have a good run in the years that I've played here but I enjoyed it a lot today. I missed last year and felt really good playing here.
"I struggled a little bit in the first set with his game but every set I've been playing a bit better and in the end I was at a good level."
Gasquet looked like he had just taken a dip in the nearby Yarra river for much of the first set as Alcaraz made him sweat for points but he held firm and drew level at 3-3 in the tiebreak with a breathtaking backhand winner.
However, Alcaraz raised his level from there to go a set up and the floodgates opened as the Wimbledon champion racked up a 3-0 lead in the next before doubling his advantage in the match, showcasing a mix of explosive forehands and deft shots.
With Gasquet fading, Alcaraz stepped up another gear on a balmy evening on Rod Laver Arena for the early break in the third set and settled the contest on serve, finishing with a fiery ace out wide on his fifth match point.
