Head's seventh test hundred, the second at his home ground, helped Australia post 283 all out for a handy lead of 95 in the low-scoring contest.
Hazlewood then produced a four-wicket burst to rock West Indies, who finished day two on a precarious 73-6, still 22 behind and with a defeat inside three days looming large.
Justin Greaves fell for 24 to the final delivery of the day leaving Joshua Da Silva on 17 at the other end.
"Feels like the seam movement is sharper than yesterday, more pace in the wicket. So yes, it's nice," Hazlewood said.
"You are always in the game even if you are hit for a four occasionally."
After Australia resumed on 59-2, West Indies made frequent inroads to stay in the contest.
Occupying the number four position after Steve Smith moved up to the top of the order, Cameron Green (14) could not impress immediately.
Shamar Joseph (5-94) dismissed Green caught behind, and fellow debutant Greaves sent back Usman Khawaja (45), who was snared in the slip.
Mitchell Marsh and Alex Carey did not last long but Head counter-attacked and brought up his hundred with a driven four off spinner Gudakesh Motie before celebrating the milestone in style.
The left-hander, who hit Shamar Joseph for back-to-back sixes, went past 3,000 test runs during his knock.
Kavem Hodge took a well-judged skyer near the rope to dismiss Head, who had gone after Alzarri Joseph.
Shamar Joseph dismissed Nathan Lyon (24) to complete his five-wicket haul but world test champions Australia remained in the ascendancy when West Indies came out to bat for the second time in two days.
Hazlewood dismissed Tagenarine Chanderpaul for a duck with his first delivery and, in his next over, sent back West Indies captain Kraigg Brathwaite (one) with Head taking a sharp catch at short leg.
West Indies slumped to seven for three after Alick Athanaze fell for a duck, attempting a hook against Hazlewood only to glove the ball to wicketkeeper Carey.
Hazlewood did not concede a single run until his fifth over, when he struck again, this time Hodge edging him to Smith at second slip.
Replacing Hazlewood in the attack, Green dismissed Kirk McKenzie (26) in his first over to reduce West Indies to 40-5.
West Indies, who have not won a test Down Under since 1997, are without all-rounders Jason Holder and Kyle Mayers, who have opted to play in lucrative Twenty20 leagues instead.
