The fifth seeded Japanese, rated a strong contender to capture his maiden grand slam this year after a breakout 2014, broke Almagro six times and sealed the match with a big serve in just over two hours.
The U.S. Open finalist's victory avenged his loss to the Spaniard in the quarter-finals of his home tournament in Tokyo in 2013.
Nishikori will play the winner of Brazil's Joao Souza and Croatian Ivan Dodig.
(Reporting by Ian Ransom; Editing by Peter Rutherford)
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