Australia was bowled out for 399 but the visiting bowlers reduced the West Indies to 35-3 by the break.
Off-spinner Nathan Lyon led the way with 2-1 while left-arm seamer Mitchell Starc grabbed 1-3.
Shane Dowrich (11) and Shai Hope were the unbeaten batsmen. Earlier, Smith stretched his ninth test century to within one of a maiden double before Taylor yorked him and gained a clear lbw verdict.
Smith, overnight 135, got solid assistance from Josh Hazlewood (24) as the last three wickets added 93 runs.
His ninth wicket stand with the left-handed Hazlewood was worth 63 when Taylor produced a wicked inswinger to deny Smith, who failed in his use of the TV review.
Hazlewood, who hit two fours off 58 deliveries, eventually fell to a catch in the deep off left-arm spinner Veerasammy Permaul.
Rajendra Chandrika, on debut, was dismissed for a duck, edging behind off a Starc delivery that was perilously close to being a no ball.
Then Lyon bowled Kraigg Brathwaite (4) as he played inside the line and added a clear lbw verdict against Darren Bravo (14) to put Australia well on top.
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