The 19-year-old has found wins hard to come by since her extraordinary success in New York in September but, back on the big stage, Raducanu came up with the right formula again to claim a 6-0 2-6 6-1 victory.
She lost just four points in a startling first set that admittedly was woeful from Stephens before the American hit her stride in the second.
That set up a first ever decider at this level for Raducanu, and she could not have handled it any better, keeping her cool and striking the ball with confidence to clinch victory after an hour and 45 minutes.
Raducanu, who will take on Danka Kovinic next, said: “Both me and Sloane really put everything out there. I thought it was a really high-quality match with lots of great rallies so I'm really happy to have come through against such a great champion.
"It was a tough match-up for a first round. Her athleticism is really up there. I was having to work extremely hard for my points but I'm so happy to have got through.
"I was very pleased.
"Coming out for the first set, I thought I played some very good tennis. Of course there was going to be some adversity.
"I'm happy to have regrouped. In the third I don't think the score really reflected it. I was really feeling it."
Meanwhile, Heather Watson rode her "positive wave" to a first grand slam victory for a year at the Australian Open.
The British number two broke a seven-match losing streak dating back to August in Adelaide last week and said after a 6-3 5-7 6-2 victory over Mayar Sherif that she had rediscovered her love for the game during a pre-season training block in Florida.
"I'm really happy to get through this one because the last couple years have not been good for me," Watson said.
"I had a great pre-season in Florida and I found my love and joy for tennis and competing again. I feel like the last couple years I felt really anxious before matches, just really not even wanting to go out there, but trying to get over it because I know in sport there's always ups and downs.
"Today I was excited to play. I couldn't wait to get out on the court."
Watson will now get another crack at 29th seed Tamara Zidansek, who she lost narrowly to in Adelaide.