Australia won the first Test by 159 runs but were bowled out for 180 on day one and the top order struggled again in the second innings. They slipped to 65-4 before Travis Head, Webster and Carey bailed them out.
In Grenada teenage opener Sam Konstas and veteran partner Usman Khawaja put on 47 before both were dismissed without another run added.
Smith top-edged a pull shot to fine leg, trying to take the attack to fast bowler Joseph, as Australia lost three wickets for three runs.
Smith, who has replaced Josh Inglis, missed the first Test after dislocating his right little finger during defeat by South Africa in the World Test Championship final at Lord’s last month.
Cameron Green fell to Jayden Seales off the final ball before lunch to leave the tourists in trouble at 93-4.
Following a brief delay for rain, Head was dismissed for 29, caught behind off Shamar Joseph, with the third umpire ruling wicketkeeper Shai Hope’s take low to his left was clean.
All-rounder Webster and wicketkeeper Carey responded brilliantly to steer Australia to 209-5 at tea.
Carey was more scratchy and was dropped on 46 by Hope but also hit 10 fours and one six in his 63 off 81 balls, before he tamely picked out mid-wicket off Justin Greaves.
Webster was firm in defence in making 60 off 115 but misjudged a risky second run to deep point and was run out by Keacy Carty as Australia’s lower order subsided.