Nishikori romped through the first set in just 28 minutes against Klizan. The Japanese star will face either David Ferrer or Pablo Andujar in Sunday's final.
Novak Djokovic and Andy Murray, meanwhile, set up a semifinal showdown with convincing wins over two young players enjoying breakthrough seasons.
Seven-time champion Rafael Nadal reached the French Open third round on Friday with a 4-6, 6-3, 6-3, 6-3 win over Martin Klizan of Slovakia.
Beside Nadal, third-round matches involving seeds including number three Tomas Berdych, number five Milos Raonic and number six Kei Nishikori were all postponed.
Bhupathi and Nestor beat Romanian Victor Hanescu and Slovak Martin Klizan with a 6-1, 7-6 (8), while Bopanna and Ram lost to Italians Simone Bolelli and Fabio Fagnini with 2-6, 6-7 (3).
Roger Federer set a new record of 234 Grand Slam match wins on Wednesday when he overcame a mid-match wobble to beat Romania's Adrian Ungur 6-3, 6-2, 6-7 (6/8), 6-3 and reach the French Open third round.