The American seventh seed beat compatriot Collins 6-4 6-4 and will face Kazakhstan's 21st seed Elena Rybakina next.
Williams, 39, is one of just four top 10 seeds left in the women's draw after Aryna Sabalenka lost earlier on Friday.
The Belarusian third seed was beaten 6-4 2-6 6-0 by Anastasia Pavlyuchenkova.
Williams is bidding to equal Margaret Court's record of 24 Grand Slam triumphs, but has not made it past the fourth round at Roland Garros since 2016.
"That felt really good for me," said Williams. "Things were not going my way. I had to earn it and turn it around.
"That was really positive for me going into the next match."