The breakthrough came during talks at the Labour Relations Commission (LRC).
It followed 12 hours of negotiations and after Bus Eireann had tabled fresh proposals on reducing costs.
The Labour Court is to hear the case next week, and the company expects all services to operate as normal tomorrow and Monday.
Transport Minister Leo Varadkar welcomed the development. "I hope that both sides will be able to reach a final resolution in the near future, in the interests of passengers and taxpayers," he said.
A strike had looked inevitable until Kevin Foley, the LRC's director of conciliation, invited both sides to yesterday's talks.
Siptu voted on Wednesday to strike. Members of the National Bus and Rail Union (NBRU) and the Transport Salaried Staffs Association (TSSA) had also agreed to stoppages.
Workers were balloted after Bus Eireann management wrote directly to members of staff to tell them the company would be implementing money-saving measures starting tomorrow.
Among those who would have been hit were people living in commuter towns around Dublin.
The bus company, which has 2,500 employees, wants to cut overtime, premium and shift payments, to lower leave entitlements and increase working hours for some grades.
It is seeking €20m in savings -- €11m in operations, the other €9m in payroll. Without the cuts, it will lose a projected €16m this year.