By then, the setback had snowballed into a full-blown crisis for Australia.
Woakes removed David Warner, Australia's leading scorer in the tournament, for nine and bowled Peter Handscomb who was making his World Cup debut as replacement for the injured Usman Khawaja.
Archer bowled with hostility, hitting a hopping Carey on the grill of his helmet and dislodging it from his head.
Carey braved the blow to join Smith in their dogged bid to arrest the free-fall and Australia reached the 100-mark in the 25th over before their middle order caved in.
Adil Rashid claimed 3-54, including the wickets of Carey and Marcus Stoinis, and Archer sent back Glenn Maxwell, who scored at a run-a-ball for his 22 before being deceived by a slower ball.
Smith was run out in the 48th over and Australia eventually folded with an over to go in their innings.
England have not reached the final since the 1992 World Cup and are chasing their maiden title.
The winners will take on 2015 finalists New Zealand in Sunday's final at Lord's.
(Reporting by Amlan Chakraborty in Birmingham; editing by Toby Davis)