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Top seed Murray made short work of American Sam Querrey, winning 6-4 6-2 6-4 in one hour and 59 minutes.
Evans, the world number 51, then upset Australian 27th seed Bernard Tomic with a 6-3 7-6 (7-2) 7-6 (7-3) victory.
Tim Henman and Greg Rusedski reached round four at Wimbledon 15 years ago.
It is the first time Evans has made it so far at a Grand Slam, and he next faces French 12th seed Jo-Wilfried Tsonga.
Murray, meanwhile, plays Germany's Mischa Zverev, the world number 50.
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Querrey was the man who upset then world number one Novak Djokovic at Wimbledon last summer, but Murray made sure he did not fall victim to another shock after the Serb's surprise defeat by Denis Istomin on Thursday.
With six-time champion Djokovic out, Murray is a clear favourite among many observers to finally land the third leg of the career Grand Slam.
"I don't worry about that, really," said Murray, who has lost four Melbourne finals to Djokovic.
"Obviously, if you're to get to the final, then it has an effect. A lot of the times when I've been in the final here, I've played against him. Had some tough ones."
Querrey gave the Scot plenty to think about in the early stages of their third-round match, attacking the net and hurrying the top seed, but a first-serve percentage of just 57% was not enough to keep Murray at bay.
The American missed a fleeting chance with a break point in game eight and Murray immediately took advantage, getting the break himself in the next game with a beautiful lob.
He took a firm grip on the match with a run of six out of seven games, easing through the second set with two more breaks.
There was a flurry of resistance as Querrey reeled off three straight games to lead 3-2 in the third set, but Murray once again turned up the pressure with his return to break for a fifth time on his way to a comprehensive victory.
"Sam, especially in the first set, was hitting a huge ball," added Murray, who had no problem with the ankle he turned during his previous match.
"There was a key moment at 3-4 when I saved a break point and then managed to break the next game and had the momentum after that."
British number three Evans is set to move inside the world's top 50, with 180 ranking points already secured in Melbourne, plus at least £135,000 in prize money.
After failing to convert a match point against Stan Wawrinka in the third round of last year's US Open, the 26-year-old from Solihull grabbed this opportunity with both hands.
"It was tough, Bernard is difficult," said Evans. "He is unorthodox and I found it hard at the start. I am happy to come through in three tight sets. It could have gone either way."
Evans followed the best win of his career over seventh seed Marin Cilic on Wednesday with another terrific performance, setting the pace and holding firm when under pressure late in the second and third sets.
He was broken after holding two set points on serve in the second set but recovered superbly when facing two set points two games later, then raced through the tie-break with some brilliant all-court tennis.
When Tomic threatened again late in the third set, Evans fought off another three break points with some magnificent play that even drew applause from his opponent.
There were worrying signs of possible cramp, and a brief rain shower came to the Briton's aid when serving at 5-5, 40-40, allowing him to recuperate and dominate a second tie-break to clinch the win.
Evans, who is without a clothing sponsor, has been buying his own T-shirts in Melbourne.
"I am happy with them at the minute," he said. "One shrunk in the wash so I had to change it, but I reckon they look all right."
Murray will start as a strong favourite against Zverev, but there is plenty of danger lurking on the Scot's side of the draw.
Former winner Stan Wawrinka, the US Open champion, is through to the last 16 after a 3-6 6-2 6-2 7-6 (9-7) win over Serbia's Viktor Troicki.
The Swiss, a potential semi-final opponent for Murray, will play Andreas Seppi next after the Italian beat Belgian Steve Darcis 4-6 6-4 7-6 (7-1) 7-6 (7-2).
Should Murray get past Zverev on Saturday he could face a daunting quarter-final against four-time champion Federer, seventh seed Berdych or fifth seed Kei Nishikori.
BBC tennis correspondent Russell Fuller in Melbourne
In two of his previous Grand Slams, Evans had come tantalisingly close to breaking into the fourth round - especially at the US Open last September, when he had match point to knock out eventual champion Stan Wawrinka.
Against Tomic, he looked from the first point as if he believed this was a match he was going to win. The third set in particular was very physical, as the Australian dragged Evans around the court, and yet he was still fresh enough to win the tie-break in convincing manner.
Evans will be a top-50 player for the first time after the Australian Open - some rise from the position of 772 he found himself in just 20 months ago. Perhaps hitting his mid-20s and developing a taste for the big occasion from Davis Cup ties spurred him to commit to the ceaseless dedication, and long spells away from home, required to be a top player.
Evans should be at his peak over the next four years. With help from his coach Mark Hilton, he has built the foundations to allow this not to be as good as it gets.
Dan Evans joined Andy Murray in the last 16 of the Australian Open with a brilliant performance to give Britain two men in the fourth round.