Basically, he took the stairs.
There's an "endless stair" that goes from the bottom of the Abyss to the peak of Zirag-Zigil. He fought the Balrog up there until he at last killed it on top of the mountain.
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'But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.'
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