World number one Murray, 29, was a convincing 6-3 6-0 6-2 winner against 19-year-old Russian Andrey Rublev.
Murray will face American world number 32 Sam Querrey in the third round, while Evans will take on Australian 27th seed Bernard Tomic or Victor Estrella Burgos of the Dominican Republic.
Injury scare for Murray
It took Murray just 63 minutes to race into a two-set lead against Rublev, who was making only his second Grand Slam appearance.
But there were concerns for the Scot at 1-1 in the third set when he rolled his right ankle and cried out in pain as he fell to the floor.
A tournament doctor came out to check on Murray, who expressed his discomfort towards coach Ivan Lendl in the players’ box.
But he was soon moving more freely, although still with the occasional grimace, to wrap up the match and stay in contention for his first Australian Open title.
“It’s just a little sore. It’s not too serious,” he said afterwards.
“I definitely rolled it a bit and I’m sure I’ll get some ice on it tonight. I was moving OK. I can put weight on.
“Sam Querrey in the next round will be a tough one. He’s got a big game, a big serve and takes chances with his forehand.”