The bill says a write-in vote is valid even if a voter misspells a candidate's name - an issue in last year's race.
Murkowski's GOP rival Joe Miller sued over the state's handling of the election and it's counting of ballots for Murkowski.
Three courts refused to overturn results favoring Murkowski, who made her outsider run after losing the Republican primary to Miller.
A federal judge, however, said he could see both sides of the debate, calling Alaska's write-in law "poorly drafted." That decision was seen by some lawmakers as a call to action.
The bill passed by the Senate Monday now goes to the House, where the chair of the House State Affairs Committee, Rep. Bob Lynn, R-Anchorage, has said he wants to wait on moving legislation related to election changes until the state releases an internal review of its handling of the race.
The lieutenant governor's office, Division of Elections and Department of Law are conducting the review. Public comment closes Friday.
Review findings, which could include recommended legislation or regulatory changes, are expected by March 4.