Greater Manchester Mayor Andy Burnham said the intervention was needed as he claimed there were no signs of improvement in services despite repeated calls for action from Transport Secretary Chris Grayling.
In a letter to Theresa May, he wrote that performance on Northern Rail services 'continued to be poor' following Mr Grayling's statement in May that the issue was the number one priority for his department.
It came as a report by the Northern Powerhouse Partnership (NPP) revealed a major impact on businesses, commuters and families, with former chancellor George Osborne calling for powers including spending to be devolved to Transport for the North.
It said that businesses had lost almost Â£38 million because of Northern Rail disruption, with the cost up to Â£1.3 million a day at its worst.
And some Trans Pennine routes had seen half of services cancelled or seriously delayed on the worst days.
The full cost to the north is likely to be considerably higher, as while Northern Rail provided figures for its affected services, Trans Pennine did not.
Over the entire period, using Northern Rail figures, 945,180 hours were lost to delays, an average of 22,504 per day.
Mr Burnham pointed out that in the first three weeks of its emergency timetable, introduced in June, services in its central region arriving on time had declined to 77.2 per cent, compared to 88.4 per cent in the corresponding period last year.
Lancashire/Cumbria inter-urban services had 1,179 full or part cancellations, while Merseyside services into Manchester/Wigan and North Manchester services totalled 991 train cancellations.
The mayor said passengers were also "left in the lurch" as recently as Sunday when Northern emailed customers to inform them it expected about 47 services to be withdrawn, including from Liverpool to Manchester Airport, to ensure its trains are in the right depots for Monday, when it hopes to reintroduce most of its May 2018 timetabled services.
He said: "It is frankly outrageous for emails to be dispatched at 9pm on a Saturday night telling people that there will be a much-reduced service the following day. People heading to Manchester Airport to go on holiday will have been left stranded as will others with work and family commitments.
"Passengers cannot be left in the lurch like this. This is no way to run a railway and we cannot continue to put up with a rail service provided when the operators can be bothered. People's lives are being badly affected by this chaos and the Government cannot continue to turn a blind eye to the plight of Northern commuters.
"There are only so many times that I can call on Chris Grayling to do his job and help rail passengers in the north. He has failed to deliver on his promise to make sorting out rail chaos here his top priority and that is why I feel have no choice but to ask the Prime Minister to intervene."
In his letter to Mrs May, Mr Burnham invited her to visit Manchester "at your earliest opportunity" to meet commuters affected.
"I also urge you to call an urgent meeting with the rail industry to get clear answers as to why things are still so bad and an action plan to turn the situation around," he continued.
"Your Government cannot continue to turn a blind eye to the plight of Northern commuters. I hope that you will now take a lead in ensuring that people in the north get the rail service they deserve."
Last week the leaders of Trafford Council and Tameside Council called for the sacking of Mr Grayling.
The NPP report, Devolving our Railways, found that people had lost their jobs and missed out on opportunities to get one, with businesses suffering staff shortages and a drop in productivity.
It levelled criticism at rail firms but also at the Department for Transport and Network Rail, with the latter's overrunning engineering work sparking the initial problems.
Former chancellor George Osborne, chairman of the NPP, said: "This report sets out a clear message to government - the Northern Powerhouse is crying out for more devolution.
"This is the very role we envisaged Transport for the North doing when we set them up; Northern leaders will support them in making sure the people of the Northern Powerhouse get the modern, connected network they deserve."