Smith, 59, will serve at least one year, until January 2019, the governor announced. A special election will be held to permanently replace Franken, who last week said he'd resign amid sexual misconduct claims by several women.
"I accept this appointment, and it will be my great honor to serve Minnesota as United States Senator," Smith said. "Though I never anticipated this moment, I am resolved to do everything I can to move Minnesota forward. I will be a fierce advocate in the United States Senate for economic opportunity and fairness for all Minnesotans."
Franken's term wasn't set to expire until 2020. His actual departure date hasn't been announced.
"Tina Smith is a person of the highest integrity and ability," Dayton said. "There is no one I trust more to assume the responsibilities of this important office. I know that she will be a superb senator, representing the best interests of our state and our citizens."
In 2014, Smith was elected to serve as the state's 48th lieutenant governor. She mainly focused on "building an economy that works for all Minnesotans." She became Dayton's chief of staff in 2011 and previously was chief of staff for Minneapolis Mayor R.T. Rybak.
Smith, who received an undergraduate degree from Stanford University and an MBA from Dartmouth College, moved to Minnesota in 1984 for a job at General Mills. She was born in New Mexico.
The special election in November 2018 coincides with a re-election bid by Minnesota's other senator, Democrat Amy Klobuchar.
"I will run in that election, and I will do my best to earn Minnesotans' support," she said. "I believe the way to do that is by being the best Senator I can be.