Wayne Rooney came off the bench to add to his Manchester United teammate's early opener, and the Three Lions held on after a late Eric Dier own goal to make it two wins from two in their Euro 2016 warm-up matches.
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Manager Roy Hodgson opted to hand an international debut to Rashford in England's last match before the 23-man squad for this summer's tournament is chosen, with the striker winning his first cap just 92 days after making his senior club bow.
The 18-year-old scored on both his club debut and Premier League debut for United, and it took him just 135 seconds to add his England debut to that list as his meteoric rise continued with the opening goal of the contest in just his 19th senior appearance.
After Fraser Forster had survived a scare at one end of the field, England quickly broke up the other and Raheem Sterling's cross deflected into the air for Rashford to fire a first-time volley past Mat Ryan and make a dream start to his international career.
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The goal saw Rashford break Tommy Lawton's long-standing record as England's youngest ever scorer on their senior debut, while the United man also put himself third on the list of youngest players to have ever found the back of the net for the Three Lions, behind only Rooney and Michael Owen.
England also opened the scoring inside three minutes of their recent friendly with Turkey, but they failed to kick on in that match and the same was true this evening as Australia responded well, with Robbie Kruse powering a long-range strike over the crossbar.
Rashford had a chance to double his tally for the night after just 15 minutes when another break saw Sterling release him through on goal, but this time he got his feet in a muddle and the ball ran away from him when a better touch would have left him with only the keeper to beat.
Tom Rogic had another sight of goal for the visitors that he dragged wide shortly afterwards, but clear chances were hard to come by for both sides as the first half dragged on.
England almost created one on the half-hour mark when Ryan Bertrand pulled the back for Rashford, who in turn clipped a clever pass in to Adam Lallana, but Ryan quickly smothered the Liverpool midfielder's effort as he tried to shoot on the turn.
Lallana's club teammate was the next to threaten as Jordan Henderson, making his first start since early April, curled a first-time effort towards the top corner from the edge of the box that Ryan had to claw over the crossbar.
There were no further goals before half time, and Hodgson went for experience at the break as Rooney and James Milner were brought on to replace Lallana and Jack Wilshere, switching formation to a 4-3-3 in the process.
Rooney lined up in the central striking role, and it took the captain less than 10 minutes to make an impact when he collected Sterling's pass before lashing it into the roof of the net.
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It was Rooney's 52nd goal for his country, extending his all-time record against the same opponents who he faced on his debut, but it was the first time that he had scored for England after coming off the bench.
Rooney had another opening shortly after the hour mark when he linked up with club teammate Rashford, but this time the skipper could not connect cleanly with his finish and a presentable chance went begging.
Shooting opportunities were beginning to arrive more often for England, and Nathaniel Clyne was the next to go close with a powerful strike that dipped a yard or so over the crossbar.
Australia responded with a couple of chances of their own when Christopher Ikonomidis sent a header off target before Kruse was denied by Forster from a tight angle - the England keeper's first meaningful save of the game.
The hosts went straight up the other end and almost killed the game off once and for all as the contest began to open up, but Milner's low strike zipped narrowly past the far post after he had linked up with Ross Barkley.
Dier was introduced as a 73rd-minute sub, and it took him even less time than Rooney to make a goalscoring contribution off the bench, although this time it came at the wrong end of the field.
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Deployed at centre-back rather than the defensive-midfield position that he has played all season for Tottenham Hotspur, the 22-year-old sent a stooping header into his own net when trying to clear Milos Degenek's low cross.
Australia could not muster a comeback in the closing stages, however, as England comfortably saw out a second consecutive 2-1 win that will give Hodgson plenty to think about ahead of the May 31 deadline to finalise his Euro 2016 squad.
The Three Lions will next take on Portugal in their final Euro 2016 warm-up match on June 2 before their campaign gets underway against Russia on June 11.