The Olympic Gold medallist brushed aside the challenge of Portugal's Joao Sousa, 23, to advance to the next round, where he will face Lithuanian Ricardas Berankis.
On the hottest day of the tournament so far, with temperatures soaring above 100˚F, Andy moved quickly to conserve energy.
He saw off Sousa, 6-2, 6-2, 6-4, in one hour 41 minutes, pushing forward for a second successive grand slam following his maiden major win at last year's US Open.
Andy, 25, from Dunblane, said: "It doesn't matter how much training you do, it's tough in these conditions.
"It's extremely hot, especially when the sun comes through the clouds. It's good to get it done in three sets.
"The longer the rallies go, the tougher it becomes on the legs.
"You need to get in control of the points and dictate them because otherwise you will get very tired very quickly."
Cheered on by his girlfriend Kim Sears, and crowds of enthusiastic and noisy ex-pat Scots fans, the British number one cruised through the match.
He will now meet former junior number one Ricardas Berankis, 22.
Andy said: "I have practised with Berankis a lot.
"He is a very hard worker, it will be tough."