How to defeat Bowser the first time you battle him in Super Mario RPG.

Don't Target Bowser

In this battle, don't bother trying to hit Bowser. Instead, you should target the Kinklink behind Bowser. To quickly end the battle, use Special on the Kinklink, instead of your regular attack. When you use Special, be sure to press the button again just before Mario's feet hit the enemy, to increase the damage. If Bowser attacks, press the attack button just before Bowser's attack hits you, to reduce the damage of his attack.

After the battle, there will be a cutscene. Afterward, Toad meets you at your house.