Pavlo Lapshyn, 25, admitted stabbing to death 82-year-old Muhammad Saleem as he was walking home in Birmingham in April.
Police told the London court that Lapshyn said he murdered Saleem "because he wasn't white."
Lapshyn had reportedly arrived in Britain just days before the attack on Saleem. Weeks later, he planted bombs that exploded near mosques in Walsall, Wolverhampton, and Tipton in the west Midlands.
No one was injured in the blasts.
Police said that chemicals and bomb-making equipment were found at Lapshyn's home following his arrest in July. Lapshyn will be sentenced on October 25.