With matches against the West Indies and New Zealand to come in Group C, Jonathan Trott's side were quick out of the blocks in Guyana.
Rahmanullah Gurbaz (76) and Ibrahim Zadran (70) led the way with an impressive opening stand of 154 after 14.3.overs.
However, Afghanistan collapsed somewhat later on, adding just 29 more runs as they finished at 183-5.
Nevertheless, they were quick to ensure Uganda were unable to build any momentum. Fazalhaq Farooqi dismissed Ronak Patel and Roger Mukasa in consecutive deliveries during the first over.
Indeed, it was a true baptism of fire for the T20 World Cup debutants, who were bowled out for just 58 with four overs still remaining.
Data Debrief
Afghanistan set the tone with a fast start in Georgetown. Zadran and Gurbaz's opening stand of 154 was the second-highest in T20 World Cup history, behind only the 170 of England's Jos Buttler and Alex Hales against India two years ago.
Farooqi then took centre stage in the field, claiming his first five-wicket haul in T20Is and conceding just nine runs - four of which came from his first ball. His 5-9 was the fourth-highest figure in tournament history.
The three-time champions, targeting 292 for victory, slumped to 91 for seven before turning things around thanks to Maxwell’s breathtaking unbeaten 201, which came off 128 balls and included 21 fours and 10 sixes.
It was with the last of those sixes that the 35-year-old all-rounder – who battled on despite appearing in some pain – wrapped up the win with 19 deliveries to go.
It was the highest-ever score by an Australian in a one-day international, as the team achieved the highest successful ODI run chase there had ever been at the Wankhede Stadium.
The ground had seen history made earlier with Afghanistan recording their maiden World Cup century, Ibrahim Zadran posting an unbeaten 12.
Their total of 291 for five also featured contributions of 35 not out from Rashid Khan and 30 from Rahmat Shah as Afghanistan sought to beat Australia for the first time at the fourth attempt in this format.
Australia’s reply started badly with Travis Head being dismissed for a duck by Naveen-ul-Haq with only four runs on the board.
And the wickets continued to go as Pat Cummins’ men stumbled to 49 for four after Azmatullah Omarzai took the scalps of David Warner and Josh Inglis in consecutive balls.
When Mitchell Starc was ousted – despite questions over whether the ball had hit his bat – via a superb catch from wicketkeeper Ikram Alikhil to leave Australia at 91 for seven in the 19th over, Afghanistan looked to be closing in on a famous victory.
But Maxwell then took centre stage with a stunning display to push Australia to victory.
Having been dropped and survived an lbw appeal on review, he went on to register boundary after boundary, even though at times he looked to be in considerable discomfort and struggling to run.
Able to continue after receiving treatment on the field, he subsequently completed his double-century – and Australia’s win – in fitting fashion by crashing yet another maximum that took his partnership with Cummins (12) to 202.
Third-placed Australia advance into the last four, joining India and South Africa, ahead of finishing their group matches by playing Bangladesh in Pune on Saturday.
Afghanistan remain sixth, with one of them, New Zealand and Pakistan set to make the semi-finals – they face South Africa in Ahmedabad on Friday.
Defending 121, the Strikers' left-arm spinner picked up a hat-trick in the first over of the innings, dismissing Andries Gous, Dewald Brevis and Ibrahim Zadran, to leave Samp Army reeling at 1 for 3.
In his second over, he removed the opposition captain Faf du Plessis and Najibullah Zadran in space of three balls to further rattle Samp Army.
Jason Holder's 22 of 11 and Qais Ahmad's unbeaten 31 off 13 at No.10 meant Samp Army huffed and puffed to 80 for 9.
Earlier, opener Rahmanullah Gurbaz's blazing 56 off 28 set the tone for Strikers to post a competitive 121 for 5. His knock included four fours and five sixes.
In the end, Asif Ali smashed three fours in his unbeaten seven-ball knock of 17 to further lift Strikers.
Samp Army will have a second chance to make the final when they play the winner of the Eliminator between Bangla Tigers and Deccan Gladiators in Qualifier 2 later today.
At the time of publishing, the Tigers were 78-4 off seven overs batting first.
Both sides had already qualified for the Super 8s, but West Indies ensured they finished unbeaten in Group C, breaking a few records along the way.
Afghanistan had no answers to the co-hosts' dominant batting performance, with Johnson Charles getting 43 off 27 balls before Pooran plundered 98 off 53 balls, including six fours and eight sixes.
He was run out in the final over while searching for his century, but Shai Hope (25) and Rovman Powell (26) had already helped push West Insides towards the highest total of the tournament so far with 218-5.
West Indies did not let up during the Afghanistan chase, starting strongly when Akeal Hosein had Rashid Khan caught for a duck in a wicket maiden to start.
Obed McCoy was the standout though, as he took 3-14, including Ibrahim Zadran, who was starting to build some momentum with his 38, leaving Afghanistan all out for 114 with 22 balls remaining.
West Indies meet holders England in their first Super 8 match on Wednesday, while Afghanistan face India the following day.
Data Debrief: West Indies finish group in record-breaking style
West Indies' score of 218 was their highest-ever at the T20 World Cup, while they also registered the highest powerplay score with 92, beating the Netherlands' record of 91 against Ireland in 2014.
In the fourth over alone, they scored 36 runs - equalling the record for the most expansive over in men's T20s.
Pooran has scored the joint-most sixes in the tournament so far (13, along with USA's Aaron Jones), and has broken the record for the most sixes (128) in T20s for West Indies, going past Chris Gayle's 124.
His total of 98 was also the highest individual total at this World Cup, carrying him past 2000 runs in T20Is.
All in all, not a bad day for the West Indies.
