The 23-year-old Bulgarian, who blazed through the Wimbledon warm-up event two weeks ago, showed no Centre Court nerves in dispatching Saville, 20, like him a former junior champion at the All England Club.
After being knocked out in the second round in his last three appearances at Wimbledon, Dimitrov said he was delighted to move further into the tournament.
"I always wanted to do that. I wanted to do well on grass," he said. "At the moment I'm trying to adapt to each player each day."
Dimitrov has risen through the rankings to 13th this year after winning titles in Acapulco and Bucharest as well as Queen's.
He won the boys' Wimbledon title in 2008 and qualifier Saville, the second youngest man in the draw and now ranked 236 in the world, followed him in 2011.
(Reporting By Clare Lovell, editing by Pritha Sarkar)
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