The record Grand Slam champion, who is chasing his fifth Australian crown, has now made the round of 16 at Melbourne for the 13th straight year after a workmanlike 6-2, 6-2, 6-3 win in one hour 41 minutes.
It was his 71st match victory at Melbourne Park to extend his record at the Australian Open and set up a clash against either 2008 finalist Jo-Wilfried Tsonga or fellow Frenchman Gilles Simon.
It was a confident performance from Federer, who is looking to become the second man after Australian Roy Emerson to win five Australian Open singles titles.
He broke the 79th ranked Gabashvili's service five times and did not lose a serve, while hitting 35 winners with 18 unforced errors.
Russia's Teymuraz Gabashvili plays a shot during his men's singles match against Switzerland's Roger Federer on day six of the 2014 Australian Open tennis tournament in Melbourne on January 18, 2014
"It was a tough year last year and it took some time to figure out what I needed to do to feel better and to get my confidence back, because that's what I thought it was mostly," Federer said.
"For the last four to five months I feel that I'm back to normal again and at least I feel I've done the work to catch up and haven't had any setbacks, so it's very positive."
Federer said his first goal was to get through to the second week in Melbourne.