The nomination came despite concern from some party activists that Franken might have trouble replacing incumbent Republican Sen. Norm Coleman, The (Minneapolis) Star Tribune reported.
Franken -- who had previously declined to apologize for a 2000 commentary in Playboy in which he used graphic language to describe fictional sexual encounters -- issued an apology Saturday.
"It kills me that things I said and wrote sent a message ... that they can't count on me to be a champion for women, for all Minnesotans," Franken said. "I'm sorry for that. Because that's not who I am."
Franken said that as an entertainer he "wrote a lot of jokes. Some of them weren't funny. Some of them weren't appropriate. Some of them were downright offensive. I understand that."
Franken's sole rival for the nomination, Jack Nelson-Pallmeyer, withdrew as the party membership was set to endorse Franken and asked that Franken receive the nomination by acclamation, the newspaper said.
Republican state Chairman Ron Carey called Franken's apology an "11th-hour apology made out of political necessity."