He played the piano as promised.

Literal

Promise accordingly piano [obj-を] played.

約束どおりに ('as promised') is a set adverbial phrase — なお the base pattern ~どおりに ('in accordance with'), here with 約束. ピアノをひく ('play the piano') is the collocation for piano playing; other instruments use different verbs (ギターをひく same verb for string instruments, 笛を吹く for wind instruments, 太鼓を叩く for drums).