The Australians, big winners in the first Test in Hobart, grabbed three wickets in the middle session with a fifth day still open.
At tea, the tourists were 146 for four and trailing Australia by 313 runs with Jermaine Blackwood on 16 and Denesh Ramdin not out 19.
Australia dismissed Darren Bravo, Rajendra Chandrika and Marlon Samuels after lunch to take a firmer grip on proceedings.
Bravo, the first-innings top-scorer with 81, was caught behind off Peter Siddle for 21 and Chandrika's 130-ball vigil ended when he was trapped leg before wicket to James Pattinson for 37.
Blackwood has attempted to play his shots, while wicketkeeper Ramdin is searching for form after being in the batting doldrums for the last two seasons.
Australia's no-ball wicket curse had earlier continued when Bravo survived on 12 after he was given out caught behind only to be saved by a front-foot no-ball by Josh Hazlewood.
Pattinson had two wickets denied to him for overstepping on Sunday's third day, and to Australia's frustration it was the wicket-less Hazlewood's turn on the fourth day.
The Australians removed Kraigg Brathwaite in the 12th over for 31 when he was coaxed to cut spinner Nathan Lyon only to be snapped up by Smith at slip.
Smith remained 70 not out when he made the declaration at the start of the fourth day and finished 2015 with most Test runs of 1,474 at 73.70.
Only Joe Root (1,372) was in a position in England's current Test match against South Africa in Durban to overtake him in the final Test of the year.
The West Indies will need to survive a minimum of 1,080 balls to draw in Melbourne.
The tourists have yet to pass 300 in seven Test innings against Australia this year.
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