A vote on the measure had been temporarily tabled after the state Senate broke out in "chaos" debating an amendment to make exceptions in cases of rape and incest.
The ban prevents women from seeking abortions by making any abortions a felony for the doctor who performs them, making this different than the so-called "heartbeat" bills that have passed in other states recently, like Georgia.
Bobby Singleton, who has served as the minority leader of the Alabama Senate since January, told CNN's New Day he was appalled by the state legislature's passage of a bill, which if implemented would ban abortion except in cases where the mother's life was in jeopardy.
"There is a very good chance if this makes it to the Supreme Court that Roe v. Wade will be overturned", Cevallos declared on "Morning Joe".
Democratic Sen. Vivian Davis Figures questioned why supporters would not want victims of rape or incest to have an exception for a horrific act.
Democratic state Senator Linda Coleman-Madison called the Republicans hypocritical for advocating small government that ought to stay out of private matters but "now you want in my womb; I want you out". The American Civil Liberties Union is among the groups vowing to sue to stop the law, saying it violates a woman's constitutional right to an abortion. Under the bill, women seeking or undergoing abortions wouldn't be punished.
Tom McCluskey of March for Life Action said, "There's an optimism in the pro-life movement and a pessimism in the abortion movement".
The statement also said: "Alabama politicians will forever live in infamy for this vote and we will make sure that every woman knows who to hold accountable".
Those fighting the legislation stressed that women across the U.S. should know that abortion remains legal for now, despite these state actions, which are deemed to be unconstitutional.
Supporters said the bill is intentionally created to conflict with the U.S. Supreme Court's landmark 1973 Roe v. Wade decision legalizing abortion nationally, because they hope to spark court cases that might prompt the justices to revisit abortion rights.
"This bill will not take effect anytime in the near future, and abortion will remain a safe, legal medical procedure at all clinics in Alabama", the organization tweeted Tuesday night, along with a map showing clinic locations in the state.
State Sen. Rodger Smitherman, a Democrat opposed to the bill, spoke about how he and his wife faced the choice of whether or not to abort one of their children.
Senators voted 25-6 for the bill that already cleared the House of Representatives.
Courts have blocked bans on the procedure in other states, and the issue ultimately may be decided by the U.S. Supreme Court. But that would only kick in if Roe v. Wade is overturned.
Leana Wen, the president of Planned Parenthood, said: "At Planned Parenthood we will do whatever it takes to stop these unsafe bans so that our patients can continue to access the care they need, ".
Critics have promised a swift lawsuit to challenge the ban if enacted.
"There are already 14 cases nationwide in the pipeline, two of which are now at the Supreme Court of the United States", he said.