John Swinney said while he and Burnham disagree on independence, the meeting was “a welcome start” on further devolution.
“When the people of Scotland take issues into their own hands, the best outcomes will about,” Scotland’s first minister said.
“To that end, I will work constructively and collaboratively with the prime minister to devolve more powers from Westminster to Scotland – as long as those proposals drive economic growth that is felt in higher wages, better jobs, more opportunities for business to start and expand, and support with the cost of living.”
A Downing Street spokesperson said the UK government would review issues raised by the first minister but this would not include a second independence referendum in Scotland.
Rhun ap Iowerth said: “I was pleased to meet with the prime minister in Glasgow today where I once again pressed the need for Wales and the Welsh government to have the necessary tools to better serve our communities.
“I also reiterated the point that further devolution within Wales is a matter for the Senedd and only the Senedd.”
He added he welcomed the prime minister’s “step forward” on the Welsh government’s fiscal and borrowing powers and gave his “full support” in pushing for a stronger relationship with the EU.
The prime minister had “positive discussions” with the Welsh leader, Downing Street said.
A spokesperson added: “The first minister of Wales agreed to start a process to explore how each government could contribute to regional empowerment, reindustrialisation and achieving good growth in every postcode.”