The tree-replacement programme forms part of a £3.2m restoration of the park, with most of the cash coming from the Heritage Lottery Fund.
Work will also include conserving a grade II-listed bandstand, a boathouse and croquet pavilion. New wildlife areas will be planted.
The trees were identified as dead, dangerous or dying, in a report commissioned by the city council's parks and countryside service recently and the felled trees will be replaced be semi-mature specimens, yet to be chosen.
"The 25ft-tall Swedish whitebeams form an avenue that has become a prominent feature of the park," said councillor David Hardman. "But they are now in such a poor condition we have decided to cut them down and replace them with a hardier species.
"Cutting down and replacing these trees is essential both for the future of the park and the safety of the people who use it. The trees, planted around the 1930s, have come to the end of their lives and become unsteady.
"We are going to replace them with something grander and longer living. I believe this will add to the beauty of the park and leave a legacy for generations to come."
Exhibition Park was created as part of the 1870 Newcastle-upon-Tyne Improvement Act.