Members of the Rail Maritime and Transport (RMT) union had been due to walk out on Saturday and again on Christmas Eve in a row over the dismissal of a colleague.
An RMT spokesman said: "Industrial action has been called off as a satisfactory agreement has now been reached between the union and the company."
RMT members on the firm's sleeper services were also set to strike for 24 hours from the evening of December 21 and again from the evening of December 23.
The planned action was in support of Scott Lewis, who was dismissed after an incident over the type of ticket issued to a passenger.
No details of the agreement were released.
However, a ScotRail spokesman confirmed Mr Lewis would not be reinstated.
ScotRail said its services would now run as normal over the Christmas period following the announcement.
A spokesman said: "We are delighted that common sense has prevailed and the threat of industrial action has been lifted.
"It is the right decision. Our customers can look forward to normal services over Christmas and to travel to be with family and friends."
Scotland's transport minister, Keith Brown, said the news came as a great relief.
He said: "Nobody wants to see a strike on our railways, particularly at this time of year, and the travelling public have been left under a great cloud of uncertainty over the past few days about how they will be able to fulfil their plans over the festive period.
"ScotRail were working hard to come up with a contingency timetable but many people will have been worried about how to make it home for the holidays or get about to complete their Christmas shopping or make the most of the party season.
"So, I welcome this announcement from the RMT and I am sure passengers will also be pleased to hear that these strikes have been averted."