The top-seeded Alcaraz had far too much firepower and accuracy for the Vavassori, ranked 317 in the world, running out a 6-2, 6-1 winner in just over an hour.
"Today, I think I hit everything pretty well. I'm just hopeful I can keep going and be a bit better next round," said the Spaniard after the match.
Wearing a breathing aid as he recovers from illness, Alcaraz got off to a flying start, breaking in the very first game and sealing the set 6-2 as the Italian pushed a forehand wide.
The Spaniard again broke early in the second set, taking the third game with a sumptuous lob. A second break followed quickly afterwards as the Italian double-faulted.
Alcaraz wrapped up the match with another break of serve, sending a powerful forehand return to the feet of the Italian, who could only poke it wide.
"I had no ups and downs today which is good for me, something I'm working on," said Alcaraz.
Thursday's victory was a stark contrast to his first-round match when he required nearly three hours to dispatch local hero Botic van de Zandschulp.
"The first match of every tournament it's never easy to get used to the conditions, so I'm just trying to make the most of the time I spent here in the first round," he said.
Alcaraz will face compatriot Pedro Martinez in the quarter-finals on Friday after the Spaniard upset Denmark's Holger Rune in the final second-round match of the day.
Earlier on Thursday, Greece's Stefanos Tsitsipas saved a match point and battled from a set down to overcome local hope Tallon Griekspoor in a three-hour marathon.
Tsitsipas now faces a quarter-final clash with Italian qualifier Mattia Bellucci, who shocked second seed Daniil Medvedev in a marathon three-setter on Wednesday.
Also through to the quarter-finals in the bottom half of the draw was last year's runner-up Alex de Minaur, who defeated Czech teenager Jakub Mensik 6-4, 6-4.