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The pair join 33 other riders in Friday's opening round, with the top 20 after that contesting the medals.
London 2012 gold medallist Maher picked up just one time fault on Tic Tac, while Skelton had five on Big Star.
"I feel my horse is jumping better as each day goes on," said Maher.
Great Britain exited the team competition in the first round earlier this week, and Maher added: "Myself and Nick are trying to redeem ourselves.
"We really believed that we could come here and put in a good performance for the team, and things just didn't go our way. It just wasn't meant to be.
"We had to come out and keep pushing, and the deal today was that we made the cut. We both got the job done."
Ben Maher and Nick Skelton both remain in contention to become the British show jumper to win an Olympic individual medal in 44 years after qualifying for Friday's final in Rio.