Roach scored 25, Carlton Baugh Jr made 23, and Narsingh Deonarine got 21, as West Indies were dismissed for 148 in their second innings about 13 minutes before the scheduled lunch interval on the final day at Kensington Oval.
Starting the day on 71 for five, West Indies suffered an early setback, when Narsingh Deonarine was lbw for 21 playing back and across to a well-pitched delivery from Harris that straightened.
Darren Sammy joined Carlton Baugh Jr and they curbed their natural instincts, spending close to 45 minutes defying the Australian attack.
The West Indies captain was fortunate on 12, when Nathan Lyon dropped a sitter at deep mid-on off Peter Siddle, but he failed to capitalise on the chance.
Sammy was bowled for the addition of two runs, when he inside-edged a delivery from Shane Watson, tried to prevent the ball from rolling onto the stumps, and only succeeded in kicking it on.
Baugh soon followed, when he was caught at mid-off, chipped a delivery from Ben Hilfenhaus, leaving West Indies 116 for eight.
Fidel Edwards was dropped on one, when Watson failed to grab an uppish drive and spent close to 40 minutes frustrating the Aussies before he was caught at mid-off for three off Siddle, essaying an ill-advised big drive over the top.
The last wicket pair of Devendra Bishoo and Roach continued to defy the Aussies, spending close to 45 minutes together.
Roach was on 10, when Australia captain Michael Clarke dropped him at slip from Lyon’s off-spin before Harris bowled him to bring the innings to a close.
Hilfenhaus was the most successful bowler with four wickets for 27 runs from 17 overs, Harris collected three for 31 from 8.4 overs and Siddle snared two for 32 from 17 overs.
West Indies are seeking their first Test win over the Aussies in nine years, and first series victory in close to two decades.
WEST INDIES 1st Innings 449-9 declared
(S. Chanderpaul 103; D. Warner 2-45)
AUSTRALIA 1st Innings 406-9 declated
WEST INDIES 2nd Innings
(overnight 71 for five)
A. Barath b Hilfenhaus 2
K. Brathwaite c wkpr Wade b Hilfenhaus 0
K. Edwards lbw b Hilfenhaus 1
D.M. Bravo c wkpr Wade b Siddle 32
S. Chanderpaul c wkpr Wade b Harris 12
N. Deonarine lbw b Harris 21
+C. Baugh c Harris b Hilfenhaus 23
*D. Sammy b Watson 14
K. Roach b Harris 25
F. Edwards c Watson b Siddle 3
D. Bishoo not out 7
Extras (b4, lb3, nb1) 8
TOTAL (all out) 148
Fall of wickets: 1-2 (Barath), 2-3 (Brathwaite), 3-4 (K. Edwards), 4-17 (Chanderpaul), 5-67 (Bravo), 6-75 (Deonarine), 7-106 (*Sammy), 8-116 (+Baugh), 9-125 (F. Edwards)
Bowling: Hilfenhaus 17-7-27-4; Watson 12-1-30-1; Harris 8.4-2-31-3 (nb1); Siddle 17-2-32-2; Lyon 11-4-19-0; *Clarke 1-0-2-0
Position: Australia need 192 to win
Umpires: I. Gould (England), A. Hill (New Zealand)
TV umpire: M. Erasmus (South Africa)
Match referee: J. Crowe (New Zealand)
Reserve umpire: G. Brathwaite (Barbados/West Indies)