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Anderson hit his 14th and 15th aces of the match to push the third set into a tie-breaker, then won five of the final six points in the tie-breaker to claim the victory after two hours and 27 minutes.
Dimitrov, ranked 82nd, won his only prior ATP meeting with 35th-ranked Anderson last year on grass at Eastbourne and saved the only break-point the South African managed against him in this match, that coming in the first set.
Anderson advanced to a second-round match-up against either Israeli Dudi Sela or Australian Matthew Ebden.
Germany's Tobias Kamke ousted 129th-ranked South African Izak Van Der Merwe 6-3, 6-3 in Monday's only other first-round match.
Kamke, ranked 89th, advanced to a second-round match against defending champion and third seed Milos Raonic of Canada.
Raonic had a first-round bye, as did top seed Gael Monfils of France, US second seed and three-time San Jose champion Andy Roddick and Czech fourth seed Radek Stepanek, who won the 2009 San Jose crown.
