"If we prevail in this race, it will make Utah the most influential state in the union, because nothing will get through the Senate without Utah's support," said McMullin.
A former CIA operations officer, McMullin was a Republican until 2016, when Donald Trump won the party's nomination to run for president.
But this year, Utah Democrats opted not to nominate a challenger to Lee, a two-term hard-line conservative, ceding the field to McMullin's challenge.
Lee dismissed his rival on Monday night as "an opportunistic gadfly supported by the Democratic Party."
"You've refused to talk about which party you'd join," Lee told McMullin.