Dodig stunned the seventh-seeded Australian with eight aces to race into a 6-1 lead but, although the Croatian third seed found six more aces in the second set, an incensed Hewitt did enough to level the match.
And the tide would turn quickly in favour of the 2002 Wimbledon champion, who wasted no time wrapping up the third set and an ultimately dominant 1-6 6-3 6-0 win.
The 32-year-old Australian told atpworldtour.com: "At the start of the match I couldn't get into it much. He served fantastic, hit his spots every time. I didn't think he could play any better for a set and a half.
"I was just trying to hold serve and stay ahead in the second set. I played a good game at 4-3 to break. I was able to get up an early break in the third set and got a few more cheap points."
Hewitt will face John Isner in the semi-finals after the American came through a grueling hour-and-a-half against countryman James Blake.
There was little to separate the top seed from former US number one Blake in regular play with Isner's big-hitting approach grating against the shotmaking style of his Yonkers-born opponent.
Isner broke away in the first set tie-break to win it 7/1 but was pegged back to another breaker before pushing through to take a 7-6 (7/1) 7-6 (7/5) victory.