Throughout the primary, the 30-year-old Salt Lake City resident has given scant attention to her potentially historic status. She instead has campaigned on a progressive platform and promised to aggressively challenge Lee, whom she has repeatedly called "loathsome."
In unofficial primary returns, Snow had a 59.4 percent to 40.6 percent lead over Jonathan Swinton.
A marriage therapist, Swinton is 35 years old and Mormon. He described himself as a conservative Democrat who sought to govern as a centrist, similar to former Rep. Jim Matheson and Doug Owens, who is running against Rep. Mia Love, R-Utah, for the second time.
Snow jumped in the race shortly before the filing deadline because she wanted to offer an alternative to Swinton. And while Swinton led in the state convention, Snow notched enough support to force a primary, largely by criticizing Swinton for advocating for limits to abortion rights.
It appears Utah Democrats were similarly eager to try a different approach against Lee, one of the nation's most conservative senators. Lee faced no challenge from within his party as he seeks a second six-year term.
Swinton said the results were "discouraging."
"We hoped more Democrats were really looking at the long game at this, trying to unseat Mike Lee," he said. "The reality is I've done my absolute best and run an honorable campaign."
Snow credited her primary election performance to her focus on "issues that Democrats care about."
She has called for a $15 per hour minimum wage, paid family leave, legalized marijuana, criminal-justice reform and free or reduced tuition for higher education, a platform inspired by presidential candidate Sen. Bernie Sanders, I-Vt. She said her goal is to boost working-class people such as herself. She's employed as a cashier at a Harmons grocery and hasn't gone to college, partly due to the cost and partly because she wasn't sure what career path she would like to take. Now she's seeking to become a federal lawmaker.
Snow began living openly as woman in October 2014 and believes being a transgender person, a rarity in U.S. politics, will drum up attention and campaign money, while her Sanders-inspired run will fire up Democrats and some independents.
A poll in early June, commissioned by The Salt Lake Tribune and the Hinckley Institute of Politics, found that Lee holds a 51 percent to 37 percent lead on Snow. National political handicappers expect Utah's Senate seat to stay in GOP hands, but Snow believes she's in a strong starting position, particularly because most voters haven't heard of her.