They will call at Dunbar, Berwick-upon-Tweed, Alnmouth and Morpeth, with one service per day stopping at Cramlington in Northumberland.
The trains will also serve Reston in the Borders when the village’s £20 million station opens in March, replacing one closed 57 years ago.
They are also due to stop at a new station at East Linton in East Lothian, which is expected to open in 2024.
TransPennine’s service will be further increased to seven trains a day between Edinburgh and Berwick from next May.
Reston will also be served by one LNER service a day when it opens, making a total of eight from May.
The move follows the shelving of planned changes to the east coast main line timetable from May, which have been postponed until at least 2023.
They would have seen just four trains a day in each direction calling at Reston, which left the Rail Action Group, East of Scotland (Rages) campaign group “aghast”.
The extra TransPennine trains will come on top of three long-distance operators’ limited stop trains between Edinburgh and Newcastle.
LNER operates every half an hour, CrossCountry ten times a day, and Lumo will increase its services from two to three on most weekdays from Sunday.
TransPennine managing director Matthew Golton said it hadn’t originally intended to increase the frequency of trains on the route.
He told The Scotsman: "Creating new services between Newcastle and Edinburgh in the way that we’ve done wasn’t in the plan back in the spring.”
However, he said the timetable consultation had shown the “strength of desire” for more trains between south east Scotland and Northumberland.
Mr Golton also said TransPennine’s trains were already travelling to a depot in Edinburgh for maintenance.
He said he had told Transport Scotland: “We think there’s an opportunity here to turn these trains into passenger trains, and design the timetable in a way that when the new station at Reston is finished in the spring and when East Linton, which is under construction, is realised, they can naturally plug into it.
"Let’s see if we can make this happen – get it underway from this weekend with a view that can we make it increase in frequency from May 2023 onwards.
"There’s a desire ultimately to get to an hourly frequency.”
Rages gave the plans a cautious welcome.
Secretary Max Eaves said: “Ultimately, it comes down to network capacity between Newcastle and Edinburgh.
“TransPennine has made it very clear to us that it is up to people to use the service.
"We understand there is a demand for train services from Reston, but whether it means that TransPennine will provide that capacity over and above the initial timetable will be closely monitored both by them and by Transport Scotland.”
Transport Scotland said ScotRail’s Edinburgh-Dunbar trains – currently five a day in each direction – would stop at East Linton when it opened.
But Mr Eaves said: “Logistically, it makes sense for TransPennine to take over ScotRail's slots and services, and call at East Linton as its first stop out of Edinburgh.”