The first minister's comments came as the Welsh government announced £157m in funding for the year, backed by the extra cash in the UK government budget.
It includes £21m for diagnostic equipment for the NHS "to help cut waiting lists" and £20m for school and college repairs.
An extra £1m will be given to the Arts Council of Wales, on top of a previous boost of £1.5m, as a resilience fund to help 60 organisations facing potential financial difficulty and protect jobs.
This does not cover the £3.5m cut from the council's budget at the start of the year.
Morgan said: "It broke our hearts to see those cuts last year, but we had to make difficult decisions and we had to prioritise the NHS."
She said the Welsh government had faced austerity cuts for 14 years from the previous UK government.
The Welsh National Orchestra cut back on performances due to funding cuts, while the Blackwood Miners' Institute has been under threat because of proposed cuts by Caerphilly council.
Chairwoman of the arts council, Maggie Russell, said: "Taking a 10.5% cut, was really tough.
"So no, it doesn't make up for all that money, but we really hope that it's the beginning of the sector being able to plan, and have a greater sense of financial stability going forward."