The Polish fourth seed needed just 61 minutes to move past Snigur, who was coming off the first tour-level main draw victory of her career on clay over Daria Kasatkina.
The Ukrainian qualifier saved four match points against Kasatkina on Wednesday, and secured the win by clinching the longest first-to-seven tiebreak at tour-level in eight years with a 15-13 scoreline.
Snigur had little fight left in her for her clash with Swiatek, who, barring a brief blip early in the second set, was in fierce form as she continues her quest for a first clay-court title since she claimed a fourth Roland Garros crown nearly two years ago.
"I'm very happy. Madrid is a special place to play, because of how special the tournament is, but also because of the conditions, so I'm happy that I adjusted well to them and was solid and just playing my game," said Swiatek, who is contesting her second tournament under the guidance of her new coach, Francisco Roig.
Swiatek, a champion in Madrid in 2024, swatted away three break points to open up a 5-0 lead inside 23 minutes and closed out the set eight minutes later after Snigur finally got on the board.
Snigur enjoyed a much stronger start to the second set, finding the rhythm on her backhand to put pressure on Swiatek, who got broken for the first time in the contest to fall behind 0-2.
The Ukrainian's advantage was short-lived, though, as Swiatek struck right back and swept the next six games to book a third-round meeting with Ann Li or Alycia Parks.
Meanwhile, Hungarian world number 63 Anna Bondar upset seventh-seeded Elina Svitolina 6-3, 6-4 to claim the first top-10 victory of her career.
Swiss 11th seed Belinda Bencic eased past Petra Marcinko 6-4, 6-2 and will next take on Russian 18th seed Diana Shnaider in round three.