Governor Ron DeSantis is expected to sign the measure, which would bar teachers from teaching these topics to children under 10 years old.
Critics say the bill will isolate LGBT youth. Proponents say it is about empowering parents on education issues.
Activists have dubbed it the Don't Say Gay bill. It is officially known as the Parental Rights in Education Bill.
The Republican-backed legislation passed on Tuesday. It prohibits any instruction about sexual orientation or gender identity between kindergarten and third grade - when students are roughly between five and nine years old. It also calls on school districts to avoid LGBT topics "when not age-appropriate or developmentally appropriate for students".
State laws that ban or constrain the discussion of LGBT life in classrooms - sometimes called "no promo homo" laws by critics - are not uncommon.
Four US states - Louisiana, Mississippi, Oklahoma and Texas - currently have laws on the books that explicitly prohibit or limit sex education to heterosexual activity.
The Florida bill also extends to student support services, including counselling, and grants parents the power to sue schools directly if they believe an educator has broken the law.
Representative Joe Harding, who sponsored the bill in Florida's statehouse, has said that the bill will is simply about "empowering parents".
Critics say that vague language in the law could result in teachers and students being prevented from discussing their lives even outside the classroom.
"This is really about making it impossible for LGBT young people to speak for themselves," said Nadine Smith, the executive director of advocacy group Equality Florida.
Todd and Jeff Delmay - one of the first same-sex couples to get married in Florida - say this bill would make it impossible for their 11-year-old son Blake to talk freely about his dads.
"We have fought and won these battles to adopt children and to be a family and to be together," Todd told the BBC. "And they are now trying to find every way in which they can erase us and our families."
And the couple warned the bill could marginalise LGBT youth as they come to terms with their identity.
"A child's need for acceptance in a safe space should always take precedence over some parents' belief," Todd said.
The Florida Family Policy Council, a conservative group that supports the bill, said in a statement last week that it was "necessary because government schools in the US have become ideological, political and are more interested in shaping a child's politics and sexual inclinations than they are in teaching academics of reading, writing, math, and education".
Gov DeSantis - widely regarded as a possible 2024 White House contender - endorsed the bill last month, saying it was part of a wider effort to make the education system "transparent" to parents.
The Florida bill is part of a fierce debate taking place across the US over children's education, how schools operate and what they teach.