Seamer Josh Tongue led England's battling rearguard with 3-42 but Alex Carey got Australia across the line with a four through the covers in mid afternoon in the company of Cameron Green.
"It's nice to finish on a high note," said stand-in Australia captain Steve Smith.
"We know the importance of every test match with the World Test Championship, so to get the result here and finish the series on a high was incredibly pleasing."
England will take credit for making a game of it and not being ground into the Sydney dirt as previous tourists have in end-of-series dead rubbers at the famous old ground.
Without the bowling of injured captain Ben Stokes, however, defending 160 was always going to be an uphill challenge on a good wicket which allowed the second day-five action of the series.
Australia's openers managed to get the target under 100 runs before Man of the Match Travis Head, whose third century of the series underpinned his team's first-innings 567, ballooned a shot to midwicket off Tongue for 29.
Jake Weatherald followed for 34, again caught off Tongue's bowling, to bring up lunch with Australia 89 runs from their goal.
It was 15 years and a day since England won the final test of the 2010-11 series at the same ground by an innings and 83 runs - the last time they won an Ashes series Down Under.
Khawaja made his debut in that match and got his chance for one final innings before retirement when Smith was bowled through the gate by the spin of Will Jacks for 12 soon after lunch.
The England players formed a cordon to welcome Khawaja to the crease but Tongue's bowling was less friendly and the lefthander played on for six after facing seven balls.
"I was trying to act cool, but the whole test match I found it really hard to control my emotions," the 39-year-old admitted.
Labuschagne had been dropped on 20 off Tongue when Bethell failed to hold onto the ball despite an acrobatic leap at backward point.
He was unable to make the most of his reprieve, however, running himself out for 37 with 39 runs still required for victory.
Carey, who made 16 not out, and Green, who finished unbeaten on 22, nearly reprised Labuschagne's exit with a classic mix-up after an overthrow but eventually made their ground to secure the victory.
"It was a great game to be involved in with the test match going to the fifth day and have a bit of drama in the end," said Stokes, who exhorted his players from the slips on Thursday.
"It's a tough one to take knowing that we can play better than that. But I've got to give full credit to (Australia). They've been just incredible for five test matches."
England had resumed on 302-8 but Mitchell Starc struck a major blow to their hopes of setting Australia a testing target when he had Bethell caught behind for 154.
Bethell's superb maiden test century offered plenty of promise for the future and was essentially the difference between an innings defeat for England and Australia batting again.
Starc (3-72) returned to remove Tongue for six to end the innings on 342, the left-arm quick later awarded Player of the Series honours for his tally of 31 wickets over the five tests.
"The body is still holding together and this is a great group to be a part of," the 35-year-old said.
"I am a little tired but got the job done."
In Sydney, Australia were again that bit superior in every department, as they were when wins in Perth, Brisbane and Adelaide secured possession of the Ashes urn for another 18 months in just 11 days.
England won a lottery of a fourth test in Melbourne -- their first win in Australia since Sydney in 2011 -- but the future of the "Bazball" style of play must be in doubt after a tour that started with high hopes ended in a 4-1 loss.
"We've not won the big series that we want to be winning," said Stokes.
"When a trend is happening on a consistent basis... that's when you do need to go back and look at the drawing board and make some adjustments."
Despite England's woes, the Ashes remain as popular as ever with the 211,032 fans through the gates over the five days in Sydney, the most ever for a test match at the ground.