The fifth seed advanced 6-3, 6-1 but was made to work hard over 95 minutes, despite the deceptively one-sided scoreline.
"I knew I had to be patient, her serve is ridiculously good, so I knew I just had to hang in there and be patient," said the American, a semi-finalist at Melbourne Park in 2013.
"I thought I did that well today."
Stephens faces 31st seed Petra Martic in the third round and a potential quarter-final clash with second seed Angelique Kerber if she makes it that far.
Stephens is seeking greater consistency at Grand Slams after strong performances last year to reach the final at Roland Garros and quarters at Flushing Meadows were offset by first-round exits in Australia and Wimbledon.
Stephens and Hungary's Babos won three junior doubles Grand Slams together in 2010 and their familiarity with one another was apparent as they exchanged breaks early in the first set.
The American kept creating opportunities and was rewarded with another break to go up 5-3.
She struggled to convert her chances, requiring six break points in the next game before finally claiming the set.
Overall, she had 23 break points in the match and made only five of them count, underlining a worrying lack of killer instinct that she will need for a deep run the tournament.
There was more venom in Stephens' second set performance and she never looked back after snatching a break in the first game.
She also benefited from a poor performance from Babos, who committed 44 unforced errors and five double faults.Australian Open results
Results from Day 3 of the Australian Open on Wednesday:
Sloane Stephens (USA x5) bt Timea Babos (HUN) 6-3, 6-1
Petra Martic (CRO x31) bt Marketa Vondrousova (CZE) 6-4, 7-5
Aliaksandra Sasnovich (BLR) bt Anett Kontaveit (EST x20) 6-3, 6-3
Caroline Garcia (FRA x19) bt Zoe Hives (AUS) 6-3, 6-3
Ashleigh Barty (AUS x15) bt Wang Yafan (CHN) 6-2, 6-3
Anastasia Pavlyuchenkova (RUS) bt Kiki Bertens (NED x9) 3-6, 6-3, 6-3
Amanda Anisimova (USA) bt Lesya Tsurenko (UKR x24) 6-0, 6-2
Maria Sakkari (GRE) bt Astra Sharma (AUS) 6-1, 6-4