The 31-year-old, twice a semi-finalist at the All England Club, failed to capitalise on her early momentum after taking a 4-0 lead, and Snigur made her pay by battling back to take the opening set with little fuss.
World number 77 Snigur's flat hitting helped her break in the fifth game of the second set as the pressure began to mount on Svitolina, whose challenge appeared to fade even faster as the sun began to set over the western stands.
Snigur reeled off the games to claim her first top-10 win since 2023 in only 68 minutes and set up a meeting with French qualifier Leolia Jeanjean in the next round.
The 24-year-old jumped and punched the air after what she described as a special victory.
"Every win at Wimbledon is very special for me. This is my favourite tournament. It's amazing. It's an incredible feeling. I celebrated like I'm a football player as I watched a lot of matches at the World Cup," Snigur said.
"I was really nervous. Elina is the best in Ukraine now. It was a tough start for me. Court Two, it's a good court and one of the best here. I was nervous because there are a lot of fans here. It was tough."
Svitolina said she was not at her best against an opponent who was more comfortable on grass courts, but saw the positives despite the defeat after a long clay season.
"There were a couple of reasons that affected my performance. I wouldn't say I was at 100% as I wanted to be. Unfortunately, it's like this," Svitolina added.
"The grass is not very nice to you. It's very unpredictable in a way. When you're playing a player who is really comfortable on it and plays really good, it's just comes all together.
"It happens like this. For me, I try to look at the positive side. Now I have some time to rest, finally. After Roland Garros and the busy European swing, I didn't have much time to rest."