Saville wasted little time or energy in dispatching American teenage wildcard Clervie Ngounoue 6-0 6-2 for her first victory in New York in five years.
A one-time US Open junior champion, Saville has since struggled to produce her best tennis at Flushing Meadows, winning only three matches in seven main-draw appearances.
"When the day started I was pretty relaxed. I don't know why - it doesn't really happen on day one of a grand slam," she said of her unusually easy day at the office.
"Warming up before the match I was like 'umm, this feels like just a normal match' and that felt nice and I think that is why I played well."
Playing the Open on an injury-protected ranking after being sidelined for more than eight months recovering from a torn ACL, Saville knows she will need to find another level against Swiatek on Thursday (AEST).
"Well, the biggest goal is not to get double bagelled," said the former world No.20 who is now ranked 322nd.
"I am excited. Hopefully we'll get to play on the big court. When I was coming back from injuries that was like the goal - get to play on all the bigger courts at every slam and hopefully that will be ticked off.
"I think I can still make her life very uncomfortable. I will just play free and see what happens."
The 29-year-old is hopeful of pushing the four-time grand slam champion and says staying cool is key.
"Just play within myself," Saville said.
"Playing against anyone ranked that high the tendency for most players is to try and play their best tennis but sometimes that creates extra pressure and you miss more.
"So just playing within myself and trying to enjoy the atmosphere. I might not enjoy it if I am losing. We will see.
"I think you can definitely get in her head, she's still a human.
"I will be excited. Right now I am excited, I like big stages. I like being seen and I've played some of my best matches against top-10 players.
"I believe I can beat those players. Over 10 times I have beaten top-10 players and that gives me confidence."
It will be easier said than done, but the Australian will try to play on her terms against the top seed.
"Forehand is my weapon and, when it's working, this is when I am dangerous," Saville said.
"Obviously I have to move well to get to use my forehand but today I also served pretty well, hit my spots well. It is part of my game hunting forehands."
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