The Brazilian star was sent off in the 76th minute after receiving a second yellow card for punching the ball into the net.
Botafogo capitalised on their advantage, scoring the winning goal just 10 minutes later.
Neymar's apology
Following the match, the 33-year-old forward took to social media to express his remorse, as reported by Goal.
"Desperation to score sometimes leads us to make mistakes," Neymar wrote. "I want to apologise to my teammates and the fans! I made a mistake, forgive me!!!
"I'm sure we could have gotten the three points if I hadn't been sent off. Congratulations to the team for their performance today! Put those three points on my account!!!"
Neymar also contended that his first yellow card was unwarranted, suggesting his dismissal could have been avoided.
"Note: the second yellow was deserved, but the first one was the referee's blunder," he added. "I made a mistake and I already received a yellow... What a terrible referee. That's just my opinion... please don't punish me anymore."
Santos will be back in action on June 13, facing Fortaleza away from home. They will, however, be without the suspended Neymar.