The annual catch, in which residents of Taiji village corral hundreds of dolphins into a secluded bay and kill them, sparked renewed global criticism after US ambassador to Japan Caroline Kennedy tweeted her concern earlier this month at the "inhumaneness" of the hunt.
But Abe defended the traditional catch.
"The dolphin fishing that takes place in Taiji town is an ancient fishing practice rooted in their culture... and supports their livelihoods," he told CNN in an interview uploaded onto the broadcaster's Japanese website late on Friday.
"We hope you will understand this," he added, noting that he was aware of criticism of the hunt.
"In every country and region, there are practices and ways of living and culture that have been handed down from ancestors," the premier added. "Naturally, I feel that these should be respected."
Activists from the international militant environmental group Sea Shepherd have streamed live footage of the dolphin capture in Taiji, which caught the worldwide spotlight in 2010 when it became the subject of the Academy Award-winning documentary The Cove.
Defenders of the hunt say it is a tradition and point out that the animals it targets are not endangered, a position echoed by the Japanese government.
They say Western objections are hypocritical and ignore the vastly larger number of cows, pigs and sheep butchered to satisfy demand elsewhere.