Williams lost the opening set then saved a match point in the third before eventually prevailing 4-6 6-4 7-6 despite making 70 unforced errors.
"I tried not to let anything discourage me today," said Williams, who needed a wild card invitation just to get into the tournament.
"I kept thinking about how I really need this match and I need these points. I keep thinking about the Olympics and I need my points."
Williams, playing her first tournament since she was diagnosed last year with Sjogren's syndrome, advanced to a fourth round meeting against former French Open champion Ana Ivanovic, who beat Daniela Hantuchova 6-2 7-6.
"You have to be ready emotionally," Williams said.
"I think about how miserable I'm gonna feel when I have a loss and I still spend that same time having a win, so I just keep try to digging deeper."
Britain's Andy Murray, the fourth seed in the men's tournament, was given a free ride into the fourth round when his opponent, Canada's Milos Raonic, withdrew.
Raonic, who won the Chennai Open in January, pulled out because of an ankle strain he suffered in his second round win over French qualifier Arnaud Clement.
Murray, who won in Miami three years ago, will next play Gilles Simon of France.
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