Georgia, Kentucky, Mississippi and Ohio have enacted similar so-called heartbeat bills, while Missouri lawmakers approved an eight-week ban on abortion.
Alabama has gone even further, outlawing virtually all abortions, even in cases of rape or incest.
None of the bans has taken effect, and all are expected to face legal challenges.
Louisiana's prohibition would take hold only if neighbouring Mississippi's law is upheld by a federal appeals court.
A federal judge temporarily blocked that Mississippi law.
Abortion rights activists said Louisiana's bill would effectively eliminate abortion as an option before many women realise they are pregnant, calling the proposal unconstitutional.
The legislation includes an exception from the abortion ban to prevent the pregnant woman's death or "a serious risk of the substantial and irreversible impairment of a major bodily function" — or if the pregnancy is deemed "medically futile".
But it does not include an exception for a pregnancy caused by rape or incest.
A doctor who violates the prohibition under the bill could face a prison sentence of up to two years, with the revocation of their medical license.
Although similar abortion bans have drawn sharp criticism from Democrats nationwide, Louisiana's proposal won wide bipartisan support and was sponsored by a Democrat from the northwest corner of the state, Senator John Milkovich.
Support from Edwards, running for re-election this autumn against two Republicans, is expected to help shore up his position with some voters in his conservative home state, even if it puts him at odds with national Democratic Party leaders and donors.
- AP