The Swiss great, 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.
"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.
"I didn't have the best of Wimbledon and US Opens last year, and I didn't want to get ahead of myself," he said.
"Now the draw is getting extremely difficult, regardless of whether it is Jo-Wilfried or Gilles, but I'm looking forward to it ... I've done the work and haven't lost a set so far."
Russian third seed Maria Sharapova battled into the Australian Open fourth round on Saturday with a hard-fought, straight-sets win over France's Alize Cornet.
The four-time Grand Slam champion came through her match on Rod Laver Arena 6-1, 7-6 (8/6) in one hour 51 minutes, but Sharapova's killer instinct deserted her in the second set when she failed to serve out for the victory, forcing a tiebreak.
"She started playing better when I gave her some chances," said Sharapova, after setting up a clash with 20th-seeded Slovak Dominika Cibulkova for a place in the quarter-finals.
"After my last match I'm just glad to be through and have another day to recover."
"I definitely need to step it up -- every player will need to raise their game," she added, contemplating the tournament's second week.
Sharapova is still on the comeback trail from a shoulder injury and she was pushed hard in draining heat during her second-round match against Karin Knapp, eventually coming through a marathon third set that went 10-8.
Despite those energy-sapping conditions, she showed few ill-effects on Saturday although in temperatures that were some 20 Celsius cooler, she was still using ice vests on the changeovers.
Getting past Sharapova was always going to be a big ask for Cornet. She has only beaten a top-10 player twice in her career and had only been beyond the third round once in 31 Grand Slams, at the 2000 Australian Open.
In contrast, the Russian has reached eight major finals and won four, with her last success coming at the French Open in 2012.
The third seed got the match underway serving to love and made an early breakthrough in a tight second game highlighted by a series of entertaining baseline rallies, breaking to go 2-0 ahead when Cornet hooked a forehand long.
Sharapova's serve was finding its mark and she comfortably held before her pinpoint ball placement forced Cornet into a series of errors that allowed her to go 4-0 in front.
But her serving issues returned in the fifth game when a double fault gave Cornet a second break point and she grabbed the opportunity.
Unperturbed, the Russian broke straight back to love and served out for the set in 32 minutes.
Sharapova threw away a chance to ram home her advantage in the third game of the second set, blowing three break points as Cornet fought a rearguard action.
The Frenchwoman, who partnered Jo-Wilfried Tsonga to Hopman Cup success earlier this month, made the most of her reprieve, breaking the third seed with a deft drop shot to take a 3-1 lead.
But she wasted the advantage when she let the Russian break back.
With the finish line in sight, Sharapova got another crucial break in the seventh game with a forehand down the line to go 4-3 in front.
Cornet saved a match point on her serve at 3-5 to keep the tie alive and Sharapova then failed to serve out for the match.
She paid the penalty with the showdown going to a tiebreak, in which she saved a Corner set point before finally getting the win to book her spot in the last 16.