Double defending champion Alcaraz came through 6-3, 6-3 and will face last year's runner-up Jan-Lennard Struff in the next round after another impressive showing before adoring fans on home soil.
World number three Alcaraz missed a month prior to Madrid with a forearm issue but returned to crush Alexander Shevchenko in the second round and looks in fine shape ahead of the French Open, the second Grand Slam event of the season.
Still wearing a protective sleeve on his right arm, he had far too much for Seyboth Wild, who had ousted 28th seed Lorenzo Musetti.
"Maybe today I was a bit nervous about how it would be, Thiago hits the ball so hard, I didn't know how the forearm would hold up," said Alcaraz on court.
"Today was the test of fire, no discomfort, feeling spectacular... from here, we'll be getting better."
After taking two breaks in a solid first set, Alcaraz stepped up a gear in the second, breaking in the second and fourth games as he raced into a 5-0 lead.
Seyboth Wild battled to a hold and a break to salvage some pride but Alcaraz triumphed on serve when the Brazilian went long.
"I was very comfortable on the court, moving, hitting the ball, as I always say finishing the game isn't easy, it's nerves," added Alcaraz.
Germany's Struff defeated Frenchman Ugo Humbert 7-5, 6-4.
Andrey Rublev came back from 5-0 down in a first-set tie-break to beat Alejandro Davidovich Fokina 7-6(10), 6-4 in a tense affair.
The world number eight saved five set points to stun his opponent, before a similarly hard-fought second set.
Rublev broke for a 5-4 lead with a shot which the Spaniard chose to leave as he thought it was long, but Hawk-Eye showed it was in by a whisker.
Sabalenka pushed
Women's world number two Sabalenka scraped past plucky 19-year-old Montgomery 6-1, 6-7(5), 6-4 in a two-and-a-half hour tussle.
The big-hitting Belarusian started her Madrid title defence slowly in the second round with a three-set win over Magda Linette and again was tested after claiming the first set easily.
Sabalenka broke twice for a 5-1 lead against Montgomery, who steeled herself and battled well in the final game, saving three set points before Sabalenka sealed it.
The American hit two aces to hold in the first game of the second set, showing she would not roll over for the two-time Grand Slam champion.
Sabalenka survived two set points in the gripping tie-break before 183rd-ranked Montgomery forced a decider.
The Australian Open champion broke in the second game but Montgomery hit back immediately.
Sabalenka brought up three match points in the 10th and final game, taking the first when Montgomery pushed a shot into the net.
"I'm super happy with the win, she played unbelievable tennis," said Sabalenka on court.
"I expected something like that, she's young, I saw her games and I was prepared for a great battle."
World number four Elena Rybakina eased to a 6-1, 6-4 win over Egypt's Mayar Sherif to reach the last 16.
The Kazakh, who has a tour-leading three titles this season after triumphing in Stuttgart last week, broke twice in the first set to move 5-0 ahead on the Spanish clay, before edging a closer second stanza.
Rybakina will face teenager Sara Bejlek in the next round after the Czech defeated American Ashlyn Krueger 6-3, 6-1.
Mirra Andreeva defeated Wimbledon champion Marketa Vondrousova 7-5, 6-1 to return to the last 16 for the second year running.
The Russian, who turns 17 on Monday, won 25 points to nine in the second set against the world number seven from the Czech Republic.
