It was England's first real test of the tournament after their game against France was called off due to the dangers posed of playing through a typhoon.
The Australian side put England out at the 2015 World Cup, but they have since enjoyed a seven-win streak over the Wallabies, and made it eight on Saturday.
Farrell was magnanimous in victory, praising his opponents.
"I thought Australia made that a brilliant game. They attacked throughout but our boys did well in defence and managed to get some field position off the back of it," he said.
He then went on to explain that his side new how to exploit territorial advantages, and that they would be able to cope under pressure, too.
England's centre Manu Tuilagi is tackled by Australia's fly-half Christian Lealiifano and Australia's centre Samu Kerevi during the Japan 2019 Rugby World Cup quarter-final match between England and Australia at the Oita Stadium in Oita on October 19, 201
Image credit: Getty Images
Farrell said: "We know when we have field position we can be pretty dangerous. We did what was needed. We had the lead and Australia were throwing everything at us again. We wanted to play the game at our pace and we did that in the second half."
Twice world champions Australia were losing finalists in 2015 and while their early departure will probably mark the end of Michael Cheika's five-year reign as coach, he will be pleased they went down attacking with ball in hand.
"We played an attacking style of rugby which I think really threatened the English today," said skipper Michael Hooper."Congratulations to England for a good win. We are really upset, we emptied into this and didn’t get it. We are gutted."
“Defensively, particularly in that first 20 minutes, we had to really dig in,” Jones told a news conference. “They had 70% possession and had a lot of field position so it was an important part of the game.
“We hung in there, got a bit of momentum back and then took our opportunities well.”
Eyebrows were lifted when Jones dropped George Ford from the starting team to enable him to pair centres Manu Tuilagi and Henry Slade outside Owen Farrell at flyhalf but, not for the first time, the vastly experienced coach showed that he knew what he was doing.
“We were pleased with selection at 10, 12 and 13,” he said with a smile. “They had a lot of defensive work early in the game and we thought that might happen. I also thought George Ford was absolutely spectacular when he came on – he kept Australia running around when we wanted them to.” On his 50th appearance, May was the man to finish the first of England's opportunities, one cleverly created with a series of dummy runners and the second coming after brilliant work by Slade, with prop Kyle Sinckler and Anthony Watson also scoring in the second half.
“There is probably no more professional player than him,” Jones said of May, adding that he was withdrawn late on as a precaution after feeling a “twinge” but was not expected to be a doubt for the semi-finals.
Slade’s recall was worth it almost just for that brilliantly imagined and executed chip-kick on the run into the path of May for the second try after he had claimed an interception in his own half.
Farrell also had his best game of the tournament by far, leading from the front with his aggressive defence and delivering a superlative goalkicking display, landing all eight of his kicks, several from wide out.
Jones also praised his captain's leadership, saying the team’s ability to keep their focus when Australia got back to within a point early in the second half was a big step forward.
“We gathered under the posts and we didn’t really talk about what had happened, it was about what was next,” said Farrell, who scored 20 points from four penalties and four conversions.
“We wanted to play the game at our pace, not theirs, and thankfully we did that in the second half." Having chalked up their seventh successive win against the Wallabies to fully make amends for the 2015 World Cup defeat, England now move on to Tokyo to prepare for their first semi-final since 2007, looking likely to be against New Zealand, when Jones said they will have to improve again.
"We haven’t played at our best yet and the challenge is how do we get better next week?" he said.
"The semi-final is probably the toughest game of the tournament - two teams desperate to get to the final, and everyone empties the tank."