The five-time SW19 champion was tested in the opening set by the big-serving world number 93 from Hungary, but by the second she was in full control and emerged an emphatic winner on Centre Court.
Watson, the British number one, now awaits Williams as she seeks to add the Wimbledon crown to the Australian and French Open titles she has already claimed this year.
Babos trailed 2-0 in the blink of an eye and the early break conceded as she settled into the match was to prove costly, but she held her serve in the third to gain a foothold in the opening set.
Williams challenged a call when chasing the third of a hat-trick of aces in the sixth and while she was unsuccessful, a forehand smash and another ace enabled the American to take the game with ease.
Just as Babos appeared to have settled, showing poise to win the seventh, Williams responded by speeding through the next game with a series of unstoppable serves including one that clocked 121mph.
Babos was impressive her on her own serve and it was this resistance that ensured it took Williams 35 minutes to win the set.
A long, looping forehand saw Babos broken immediately when the second set began, but Williams had to survive two break points in the fourth, greeting her eventual success with a roar.
The end arrived for Babos in the seventh as a brutally methodical Williams closed out the game without dropping a point and marked her progression beyond the second round with a celebratory twirl.