The union has not announced any stoppages and will seek an immediate meeting with the company first.
However, the Scottish Government, which nationalised ScotRail in April, said “there can be no increase on the current pay offer” and said industrial action was likely to mean “continued timetable changes and more short-term cancellations”.
RMT members, who include train conductors and station staff, voted by 1,199 to 364 for strikes – nearly 77 per cent.
They also voted by 1,336 to 228 for action short of walkouts.
A total of 1,568 of the eligible 2,257 staff voted, a turnout of 69 per cent, after the union rejected the pay offer last month that included an additional £300 payment for the use of technology.
Other staff involved include train ticket examiners, hospitality staff, engineers, cleaners and ticket office staff.
The union would have to give two weeks’ notice of any strikes, which could significantly affect services outside the Glasgow area where trains are controlled by conductors.
Most trains around Scotland’s largest city are controlled by drivers, who accepted a similar 5 per cent offer in July.
RMT Scotland organiser Mick Hogg said: “We will now be seeking an immediate meeting with ScotRail and improvement on the current offer that matches the cost-of-living crisis.
"Failure to make an improvement on the offer will result in strike action, and if that means co-ordinating action with the national dispute, then so be it.
“RMT remains available for meaningful talks.”
ScotRail said would “engage with representatives of the RMT to discuss the next steps and assess the impact of any potential disruption on customers”.
It said the rejected offer included five years’ no compulsory redundancies and an excess revenue share premium potentially worth up to £195 every four weeks if revenue targets were exceeded.
Head of customer operations Phil Campbell said: “This is a disappointing outcome given the strength of the pay offer.
"Strike action will damage the railway’s recovery at this fragile time and cost our staff through lost wages.
“We made a significantly improved pay offer to the RMT that reflects the cost-of-living challenges faced by families across the country, while balancing it against the need to provide value for the taxpayer.”
A Scottish Government spokesman said the extra £300 payment was “recognition of the need to use technology, like mobile devices and upgraded ticket printing machines.
“Any industrial action will undoubtedly have negative impacts on passengers and also employees.
"It is also likely to mean continued timetable changes and more short-term cancellations disadvantaging everyone.
"The issues at hand need to be settled but there can be no increase on the current pay offer.”
The news came as the RMT announced a further strike across Britain on Saturday, October 1 in a separate pay dispute that is likely to again significantly curtail ScotRail trains because Network Rail signalling staff are among those expected to take part.
The RMT said it also involved job security and working conditions.
Members of drivers’ union Aslef will walk out at 12 train operators, including four cross-Border firms, on the same day, and also on Wednesday 5.
The strikes, also over pay, include at LNER, Avanti West Coast, CrossCountry, TransPennine Express and will also affect travel for delegates and visitors to the Conservative party conference in Birmingham.
Edinburgh-London operator Lumo, ScotRail and Caledonian Sleeper are not involved.
Aslef members at England-based Chiltern Railways, Greater Anglia, Great Western Railway, Hull Trains, London Overground, Northern Trains, Southeastern and West Midlands Trains are also due to strike on the two dates.
However, the RMT said it would not be calling a further strike on October 5.
An Aslef strike planned for Thursday last week was called off as a mark of respect following the Queen's death, while the RMT similarly cancelled its strikes on Thursday and Saturday.
Aslef said it was in for the "long haul" as the rail disputes remain deadlocked.
RMT general secretary Mick Lynch said: “Transport workers are joining a wave of strike action on October 1st, sending a clear message to the [UK] Government and employers that working people will not accept continued attacks on pay and working conditions at a time when big business profits are at an all-time high.”
A spokesperson for the Rail Delivery Group, which represents the industry, said: "These strikes will once again hugely inconvenience the very passengers the industry needs to support its recovery from the ongoing impact of the pandemic.”