WASHINGTON — The Senate Judiciary Committee on Tuesday approved a broad overhaul of the nation’s immigration laws on a bipartisan vote, sending the most significant immigration policy changes in decades to the full Senate, where the debate is expected to begin next month.

The 13-to-5 vote came as the committee reached a deal on one of the final snags threatening the legislation — and agreed to hold off on a particularly politically charged amendment, which would have added protections for same-sex couples.

After intense behind-the-scenes negotiations, Senator Orrin G. Hatch, Republican of Utah, struck an agreement with the group of eight senators who drafted the original bill to address his concerns about visas for skilled foreign workers who could fill jobs in the high-tech industry.

By late afternoon on Tuesday, Mr. Hatch had said that he would support the bill out of committee, if not necessarily on the Senate floor, after the committee agreed, via a voice vote, to pass his amendment.