Chile were rattled as Australia's resurgence continued into the second half, but the Socceroos could not find an equaliser and Jean Beausejour made the points secure with a stoppage-time third.
Below, Sports Mole analyses whether the result was reflective of the action in Brazil.
On target: 5
On target: 4
Was the result fair?
The scoreline was certainly not as straightforward as it suggests. Chile were cruising early on, but Cahill's header gave the Aussies the belief which inspired a second-half performance in which they created more chances than the South Americans. Chile may have done just enough to win it, courtesy of their additional quality in the final third, but the goal in stoppage time put an unfair gloss on the result.
After taking the opening 10 minutes to settle into their passing game, Chile showed why many are tipping them to have a major impact at the tournament with some slick and classy offensive moves between Sanchez's opener and the half-hour mark. Cahill's equaliser showed that their defence, small in stature, is vulnerable to a strong aerial presence, while it also has a tendency to overplay. This caused its own problems, as the back four is not particularly well protected and susceptible to counter-attacks. They will need to improve defensively if they are to progress as far as some are expecting them to. At the other end it is a different story, though. They were savvy in their decision making and clinical in their finishing.
The fighting spirit of this young Socceroos side cannot be doubted, but what it is lacking is quality. Their golden age of Premier League stars are years into retirement and this is now an inexperienced side, particularly at the back, and this was evident with early moments of hesitation and weakness which cost them goals. Both full-backs were more confident going forward and often provided good delivery from the flanks, which Cahill, still as dangerous with his head as ever, thrived on. Further up the wing, Tommy Oar and Mathew Leckie had the pace which helped them get up the field, but struggled for any sort of consistency with the end product. Ange Postecoglou's side are certainly in this together, but probably not in this for long looking at the calibre of their opponents.
Sports Mole's man of the match
Alexis Sanchez: Players of the Barcelona forward's quality often decide tight games like this, and Sanchez did just that. He showed composure to put Chile ahead, and then produced a lovely turn and change of pace in the build-up to their second. He even had a role in Beausejour's late strike.
While Claudio Bravo's wayward piece of distribution did not lead directly to Cahill's goal, Chile never fully regrouped defensively and were thus short of numbers in the box when the goalscorer isolated himself against Gary Medel. It was one of the instances in which Jorge Sampaoli's side took unnecessary risks, and they were punished.
Four matches into the tournament, it is a relief to be talking positively for the first time about the performances of the officials. Noumandiez Doue and his Ivorian team were generally good, and got the big calls right, including correctly denying Cahill a second header when the Aussie strayed marginally offside.
Chile: Chile will qualify for the last 16 with victory on Wednesday, but beware of a response from their opponents Spain, who are on the back of a shock 5-1 thrashing at the hands of the Netherlands.
Australia: A daunting task in bringing Netherlands back down to earth in Porto Alegre on Wednesday. Should they lose, the Aussies would effectively be out of the tournament already.