They were nearly together at the close too, only for Smith to pick out short third man for 18 with three needed.
After extending their winning streak to nine in ODIs, the Australians next face the West Indies, who opened with a comprehensive win over Pakistan.
"It was a great win, a good start", Warner said in a post-match TV interview, admitting "there were a few nerves" getting back into the swing of things.
For five-time champion Australia, it presented a chance to raise the stakes in the reintegration of Smith and Warner following their 12-month bans imposed in the wake of a ball-tampering scandal in South Africa previous year.
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Two spectators dressed up as sandpaper stood on an overlooking balcony holding a big image of a cricket ball.
Cummins (3-40) also wasted no time, nicking off a slashing Hazratullah Zazai with one that went across the left-hander before claiming two late wickets.
Afghanistan's hopes were dashed by Finch and Warner, who braved a fine opening spell from seamer Hamid Hassan to break the back of the chase, albeit that Warner was dropped on 10 with Rahmat unable to cling on to a tough one-handed chance at slip after Hassan induced the edge. But you can just about bet it's going to keep coming, particularly when Australia play England towards the end of the tournament.
A full 434 days after Warner and Steve Smith presided over the Cape Town sandpaper plot, both men were back in polite society, marking their first official post-ban worldwide with an easy win over Afghanistan.
Much is being made of Afghanistan's outstanding spin bowling contingent, with Rashid Khan, Mujeeb Ur Rahman and Mohammad Nabi all high of the one-day worldwide bowling rankings.
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He may have been regretting that call as soon as three balls in when Mitchell Starc knocked back Mohammad Shahzad's off stump with a searing yorker before a run was on the board.
"Luckily I got the nod, we're pretty lucky that we've got three pretty good bowlers sitting on the sideline", he added.
"We expected what we were going to cop".
"They are a champion side, so you can not afford to make any mistakes against them", he said at the post-match presentation ceremony, reported Cricbuzz. A different David Warner was on show tonight, with him taking the backseat and allowing Finch to express himself.
"We will take some positive thinking", said Zadran.
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"The team has changed from four years ago, and the last couple of years we have improved very well in every department".