WASHINGTON -- Republicans cheered on the House floor as their major tax overhaul plan passed Thursday with votes to spare.

"We are in a generational defining moment for our country," said House Speaker Paul Ryan.

The Tax Cuts and Jobs Act of 2017 shrinks the number of brackets from seven to four, lowering rates and eliminating some deductions.

The independent Tax Policy Center estimates that Americans who make between $50,000-$90,000 a year would get an average tax cut in 2018 of about $800.

President Trump took a victory lap on Capitol Hill.

"The tax is going really well, thank you very much," Mr. Trump said Thursday.

He pushed for the bill's huge corporate rate cut.

But Democrats warned the GOP will pay a political price for passing a plan that primarily benefits big business and the wealthy.

The House of Representatives passed a Republican-backed tax bill on Nov. 16, 2017. CBS News

"Republicans want you to believe that their trickle-down tax break for the rich will pay for itself," said House Minority leader Nancy Pelosi. "Never has happened."

A dozen Republicans from New York, New Jersey and California voted no on Thursday because the bill eliminates the deduction for state and local taxes.

"I just have too many constituents who are going to see their taxes go up," said Rep. Lee Zeldin, R-Shirley.

But most House Republicans were ecstatic -- and they're now anxiously watching the Senate, where a separate GOP plan has gone through several rapid revisions.

South Carolina Republican Tim Scott is one of the Senate plan's authors. He said he is convinced that workers will benefit from these tax breaks their employers are getting.

"If we don't reform our business taxes, more jobs that are today in America will be in another country. So the answer is simple: Absolutely!" Scott said.

Senate Republicans are hoping to pass their bill out of the finance committee on Friday and bring it up for a full Senate vote right after Thanksgiving. Then House and Senate Republicans would have to reconcile the differences between their plans and vote again -- a process Mr. Trump is pushing them to finish by Christmas.