The match lasted an hour and 58 minutes.
Majchrzak, 30, won 6-3, 7-5, 7-5, avenging his straight-sets loss to the 29-year-old 30th-seeded Chilean in the first round of this year's French Open.
The Pole produced an impressive display on grass, dominating most aspects of the match, firing 16 aces to Tabilo's nine, hitting 43 winners compared with 33 for his opponent, and committing just 18 unforced errors against the Chilean's 27, Polish state news agency PAP reported.
Majchrzak arrived at the All England Club in strong form after capturing his first ATP title in 's-Hertogenbosch, where he defeated three top-10 players en route to the trophy.
Ranked a career-high No. 45, he will next face either American Zachary Svajda or Spain's Pablo Llamas Ruiz for a place in the third round.
Fellow Pole Hubert Hurkacz has also advanced to the second round, after defeating 11th-seeded Casper Ruud of Norway in straight sets on Monday.
Hurkacz will face Austria's Sebastian Ofner in the second round on Wednesday.
Hubert Hurkacz. Photo: EPA/NEIL HALL
On the women's side, Poland's Iga Świątek was scheduled to begin her title defence later on Tuesday against American Taylor Townsend.
A victory would set up a second-round meeting with former Wimbledon finalist Karolína Plíšková of the Czech Republic.
(gs)
Source: IAR, PAP