CHICAGO — Illinois took a crucial step on Tuesday toward ending a budget impasse that has imperiled the state’s fiscal future, but major questions remained even as a threat loomed that the state’s bond status might be lowered to junk.

In a rare holiday session in Springfield, where Democrats control both legislative houses, the Illinois Senate voted 36 to 18 to raise taxes, and then later in the day swiftly overrode the governor’s veto of the measure. The actions brought the state perhaps the closest it has come to a budget in more than two years — the longest any state has gone without one. The measure would raise $5 billion by increasing the personal income tax rate and the corporate tax rate.

The issue now comes down to a test of clashing philosophies and political loyalties after years of growing pressure: Will the minority of Republicans in the House stick with Gov. Bruce Rauner, a multimillionaire who had never held political office before, in his rejection of a tax increase without changes like a property tax freeze and cuts to workers’ compensation?

In vetoing the budget, the governor said it had failed to solve Illinois’s deep fiscal problems.

“This budget will require even more tax hikes to balance the budget and pay down the bill backlog,” Mr. Rauner said. “This budget puts Illinois on track for major future tax increases and will lead us to become the highest-taxed state in America in the coming years.”

After a hectic day of back and forths, a final test is expected to come this week when House leaders try to summon enough bipartisan support — including from more than a dozen House Republicans who have already broken with Mr. Rauner — to block the veto and move forward with a budget.

Michael J. Madigan, the House speaker and one of the most powerful Democrats in the state, told a local television station…