Rep. Tom Cole (R-Oklahoma) and Rep. Betty McCollum (D-Minnesota) are sponsoring H.R.3137.
No one spoke in opposition to a provision that seeks to strengthen an executive order issued by then-president Bill Clinton in 1996.
President Barack Obama is threatening to veto H.R.2822 due to lower funding levels and controversial policy riders.
The limited definition penalizes American Indians and Alaska Natives who aren't enrolled in a federally recognized tribe or an Alaska Native corporation.
Sen. Lisa Murkowski (R-Alaska) said the money will help tribes strengthen their justice systems.
Lobbyists for the Morongo Band of California and aides to Sen. Patrick Murphy (D-Connecticut) argued against a prior version of the rule.
The final rule does not include a controversial provision that would have allowed groups to submit new petitions.
The Menominee Nation and the Department of Justice are seeking to resolve a conflict that affects self-determination contracts nationwide.
The Senate Committee on Indian Affairs has held six hearings in a decade on a problem that affects tribes across the nation.
The move came as the Ute Tribe of Utah announced that it will join the fight against the controversial standards.
A federal judge denied a motion for a temporary restraining order but will proceed to the merits of the case.
The rule is set to go into effect on Wednesday but could be blocked by court action of Congress.