Her announcement comes three days after Leslie resigned the seat and announced her plan to run against state Sen. Greg Brower, R-Reno, in November.
Smith has represented Assembly District 30 for 10 years, and has served as Assembly speaker pro tempore and chairwoman of the Ways and Means Committee.
She said she decided to run for the Senate after consulting with family and friends the last few days, and she thinks it "is the right decision at the right time."
"I am running for the state Senate to continue fighting for a better education system, a healthier economy, and a more efficient and effective government for Nevada," she said in a statement.
Had Smith won a sixth term in the Assembly in November, it would have been her final term because of term limits.
Assembly Majority Leader Marcus Conklin, D-Las Vegas, said Smith would be missed in the lower house.
"She has been a strong leader on issues critical to our state, including improving education and making our state budget process more efficient and accountable," he said, adding she has shown she can work with lawmakers of both major parties to reach compromises on contentious issues.
During the 2011 session, Smith played a key role in crafting a balanced budget, which restored hundreds of millions of dollars in education funding that had been cut in Republican Gov. Brian Sandoval's budget.
Democrats hold an 11-10 majority in the state Senate. Leslie's decision to challenge Brower and the resignation of state Sen. Elizabeth Halseth, R-Las Vegas, on Friday will make it more difficult for Republicans to gain control.
But the GOP has pledged a major fight to win control of the Senate in November.