The world number two, who lost a set in his first-round encounter, regained his form to edge his 20-year-old British opponent with a 6-1, 7-6 (7-3), 6-2 win, after an hour and 37 minutes, reported Efe news.
On his way to the third round, the 37-year-old Swiss player recorded 10 aces and 46 winners, compared to Clarke's one and 16, respectively.
Federer is set to take on Lucas Pouille who emerged victorious in the all-French duel against Gregoire Barrere, 6-1, 7-6 (7-0), 6-4.
In other second-round action, world number seven Kei Nishikori qualified for the third round after a straight-set victory over Cameron Norrie.
The 29-year-old Japanese player defeated Norrie, world number 23, 6-4, 6-4, 6-0 in an hour and 48 minutes to move a step closer to his maiden Grand Slam title.
Nishikori fired 34 winners, compared to his rivals' 15.
He next will play Steve Johnson of the United States, who lived up to potential against Australia's Alex de Minaur 3-6, 7-6 (7-4), 6-3, 3-6, 6-3 in his second-round match.
Joao Sousa marked his 50th Grand Slam appearance with a stunning straight-set victory over 13th-seeded Marin Cilic.
The Portuguese player needed a bit more than two hours to prevail 6-4, 6-4, 6-4 over his Croatian rival.
Despite firing 10 more winners, Cilic lost the game as a result of committing 46 unforced errors and eight double errors, compared to Sousa's seven and one, respectively.
Next up in the third round, Sousa is to play Briton Daniel Evans who got past Nikoloz Basilashvili of Georgia.
Meanwhile, US' Sam Querrey emerged victorious over Andrey Rublev of Russia 6-3, 6-2, 6-3 to march toward the tournament's third round, where he is set to play John Millman.
The 30-year-old Aussie managed to advance into the third major tournament of the season at the expense of 31st-seeded Laslo Djere, 6-3, 6-2, 6-1.
Jo-Wilfried Tsonga made the round of 32 as well by beating Ricardas Berankis of Lithuania 7-6 (7-4), 6-3, 6-3.
The Frenchman is to square off against either third-seeded Spaniard Rafael Nadal or Nick Kyrgios of Australia.
( With inputs from IANS )