At stumps on the third day, England, following-on, were 203 for six -- still 129 runs behind Australia's first innings 481, which featured Smith's 143.
It looked as if left-handed opener Cook would bat until the close.
But outgoing Australia captain Michael Clarke's typically inventive decision to bring on Smith in the 77th over yielded a quick dividend when the part-time leg-spinner struck with just his sixth ball as Cook was caught off bat and pad by Adam Voges at short leg.
At the close, wicket-keeper Jos Buttler was 33 not out, with nightwatchman Mark Wood unbeaten on nought.
Also Read
England, however, at least had the consolation of having won the Ashes at 3-1 up in the five-match series.
"We're up against it, but we've just got to keep fighting and hang in there," Buttler told Sky Sports.
"The captain showed that for most of the day. Everyone's very disappointed to come here and play as we have."
Moeen Ali, whose 30 was the top score of the first innings and Wood (24), both eight not out overnight, hit out in sunny conditions ideal for batting during a ninth-wicket partnership of 57.
They fell to successive Mitchell Johnson deliveries, with England bowled out for 149.
Clarke then became the first Australia captain to enforce the follow-on in a Test since Ricky Ponting did so against New Zealand at Wellington in 2010 -- a match Australia eventually won by 10 wickets.
Opener Adam Lyth, desperate for a big score to cement his England place, fell for 10 when he edged a good length ball from Peter Siddle that cut away off the pitch to Clarke at second slip.