Federer needed just 61 minutes to seal the victory -- abetted somewhat by Dodig's six double-faults, the last of which came on match point.
The quick finish may have been a godsend for the Swiss great, who said he "tweaked" his back late in the match.
However, Federer said he fully expected to be fit enough to continue his pursuit of a first title of 2013.
"I'm not too worried," Federer said. "I have gone through it so many times where you feel a little tweak. You might play next day. Now this time around I have a day off, extra time.
"From that standpoint I'm not worried at all," added Federer, who will face compatriot Stanislas Wawrinka for a quarter-final berth.
Wawrinka ended the challenge of Australian veteran Lleyton Hewitt 6-4, 7-5.
While Federer downplayed his back trouble, Leonardo Mayer's back injury was bad enough to have the Argentine pulling out of his scheduled third-round match against Rafael Nadal, sending the fifth-seeded Spaniard into the round of 16 by a walkover.
That meant Nadal had a chance to rest his troublesome left knee, but cost him more of the match experience he says he needs as he continues his comeback from a seven-month injury layoff.
The former world No. 1, currently ranked fifth, returned in February, reaching one final and winning two titles in three clay-court tournaments in Latin America.
This is his first hard-court tournament since injury forced him out of the Miami Masters last March. Nadal remained on course for a possible quarter-final showdown with Federer. But first he faces a potentially tough fourth-round clash with red-hot Latvian Ernests Gulbis, who defeated 20th-seeded Italian Andreas Seppi 5-7, 6-3, 6-4.
Gulbis became the first qualifier to win a title this season in Delray Beach and he came through qualifying here as well. Including those qualifiers, he has won 13 competitive matches in 17 days.
The Latvian was looking forward to another crack at Nadal, who has beaten him in four prior encounters.
"I think the way I play right now, that should make a difference," Gulbis said. "Of course he's a great player, but I believe -- honestly I believe -- that if I play my best game I can beat him."
Sixth-seeded Czech Tomas Berdych also advanced with ease, beating Germany's Florian Mayer 6-4, 6-1.
Berdych lined up a clash with 10th-seeded Frenchman Richard Gasquet, a 6-1, 6-4 winner over Poland's Jerzy Janowicz.
Victoria Azarenka, the top women's seed in this combined ATP Masters and WTA tournament, got off to a slow start but turned things around for a 3-6, 6-3, 6-0 victory over Belgian Kirsten Flipkens.
Fourth-seeded German Angelique Kerber advanced on schedule, as did former US Open champion Samantha Stosur of Australia and former world No. 1 Caroline Wozniacki of Denmark.
Stosur, seeded seventh, beat China's Peng Shuai 6-3, 3-6, 6-2 while Wozniacki, a former Indian Wells winner, beat Russian Elena Vesnina 6-2, 6-1.
The women, including second-seeded Russian Maria Sharapova and No. 3 Agnieszka Radwanska who advanced on Sunday, will battle for quarter-final berths on Tuesday.