The Italian world number one is looking to beat the record of four consecutive triumphs held by all-time greats Novak Djokovic and Rafael Nadal this week in the Spanish capital.
Sinner dropped only one set on the way to trophies in Paris, Indian Wells, Miami and Monte Carlo, but Bonzi battled brilliantly to claim the first at the Manolo Santana stadium.
The Frenchman survived five break points to reach a tie-break, where he also saved set point before edging ahead.
However, Sinner stepped up a gear in the second set, securing breaks in the fourth and sixth games, and another in the third on his way to the third round.
“It was a tough match, I had some chances early on and I couldn’t use them, but this is tennis,” said four-time Grand Slam winner Sinner.
“It’s not bad to start the tournament like this, two and a half hours on court, it gives me good feedback where we need to improve for the next round.
“Of course, we need to improve, but I’m also happy, when you’re not playing your best tennis and win, it’s still a good result.”
With Sinner’s biggest rival, Carlos Alcaraz, injured and unable to compete either this week or in the upcoming French Open, the Italian is the overwhelming favourite for every competition he enters in the weeks ahead.
Earlier, Croatian qualifier Dino Prizmic sent world number six Ben Shelton crashing out with a 6-4, 6-7(4), 7-6(5) victory.
Promising 20-year-old Prizmic, ranked 87th, secured his first-ever win over a top-10 player in a three-hour battle in the Spanish capital.
American Shelton triumphed at the Munich Open on Sunday on clay but was unable to force a single break point on Prizmic’s serve.
Earlier, Italian Lorenzo Musetti, ranked ninth, defeated Hubert Hurkacz 6-4, 7-6(4).
‘I basically survived’
Women’s number two Elena Rybakina scraped into the Madrid Open third round with a 4-6, 6-3, 7-5 win over Elena-Gabriela Ruse.
Kazakhstan’s Rybakina, who won the Australian Open in January and triumphed at Stuttgart last week, struggled in the first set, making 24 unforced errors to just eight winners.
After claiming a topsy-turvy second set, she came back from a break down in the third and sealed the victory on her third match point.
Rybakina will meet Zheng Qinwen, who produced a similarly battling performance to overcome Sofia Kenin, in the third round.
“Today, I basically survived,” said Rybakina after her hard-fought victory.
“It wasn’t the greatest performance but Gabriela, she played very well. She was playing very deep and aggressive, so it was very difficult.
“My serve was not really helping me today… but I’m really happy that I managed to win, and I hope I can bring better tennis next round.”
Last year’s runner-up and the third seed Coco Gauff made light work of Leolia Jeanjean with a 6-3, 6-0 victory at the Arantxa Sanchez stadium.
Gauff will take on Romanian Sorana Cirstea in the third round, after she defeated Tyra Caterina Grant in straight sets.
