At stumps on on Sunday's third day, the hosts were 5/179 in their second innings with an overall lead of 314 runs after West End Redbacks captain Head was dismissed for a crucial 49.
The classy left-hander stepped up as third-placed WA threatened to turn the match on its head.
The visitors were behind the eight ball when their go-slow approach to their first dig finally ended when they were bowled out for 254 in 130.4 overs after resuming on Sunday at 4/165.
Trailing overall by 135 runs, the Warriors looked to have let the match slip through their fingers when SA cruised to 0/85 in their second innings.
However, Matthew Kelly (3/34) and Marcus Stoinis (1/35) combined to help WA to strike back.
Redbacks openers Jake Weatherald (49) and Henry Hunt (35) both fell with the total on 85 and it was 3-94 when the visitors claimed the prized scalp of Callum Ferguson (five).
And the the Warriors had their tails up when leading Shield run scorer Tom Cooper (four) - who hit 99 in the first dig - soon followed to make it 4/108.
But Head took on the attack, sharing a 52-run fifth-wicket stand with William Bosisto (24 not out) to settle SA nerves before he was trapped in front by David Moody late on day three.
Head thrashed six fours and two sixes in his entertaining 50-ball knock.
Chadd Sayers was SA's main destroyer with the ball with 4/54 on Sunday, rising star Wes Agar took 3/60, and Daniel Worrall took his 150th Sheffield Shield wicket for SA.
It continued an impressive game for Sayers who struck 58 not out - the highest score of his 75-match first-class career - in SA's first innings.
Earlier, Cameron Green could only add 18 runs to his overnight total to be dismissed for 43 off 149 balls with four boundaries before WA's epic first innings finally ended.
The stage was set for the visitors' snail's pace scoring rate of less than two runs an over on Saturday when Cameron Bancroft hit just 25 off 155 balls with one boundary.
WA's top scorer was Shaun Marsh who hit 63 on Saturday.