In reply to Tasmania's first-innings total of 239, WA were skittled for 155 midway through day two on Friday.
Tasmania went to stumps at 9-206 in their second innings - an overall lead of 290 with two days still left to play.
Jake Doran (32no) will try to add some more handy runs for Tasmania on Saturday when he resumes alongside No.11 Billy Stanlake (4no).
WA had a pair of plucky tailenders and Richardson to thank for staying in the match.
The Warriors were in all sorts of trouble at 9-102 on Friday before Corey Rocchiccioli (36) and No.11 Brody Couch (22no) combined for a 53-run final-wicket stand to give the home side a flicker of hope.
Richardson (4-41 off 15 overs) then weaved his magic to leave Tasmania's top-order in tatters.
Test opener Jake Weatherald (2) was caught in an ill-fated attempt to pull Richardson, and the visitors were 2-10 when Caleb Jewell (4) edged the right-arm paceman behind.
But Richardson saved his best for Jordan Silk, who had moved to 40 off 50 balls before being on the wrong end of one of the deliveries of the Shield season.
Richardson's rocket initially shaped out before swinging wildly back in to clip the top of Silk's stumps.
Silk looked back in disbelief before trudging off, with his dismissal reducing Tasmania to 4-82.
Richardson, who returned from shoulder surgery in time to feature in one Ashes Test this summer, had figures of 3-9 off six overs by that stage.
And he claimed his fourth wicket later in the innings when Aaron Hardie took a juggling catch at gully to dismiss the dangerous Beau Webster for 55.
Cameron Gannon (3-36) joined in on the fun after that to leave Tasmania nine wickets down by stumps.
Richardson took 1-29 off 16 overs in the first innings, meaning he has now bowled 31 overs for the match.
The injury-plagued 29-year-old says he's under no bowling restrictions and feeling happy to be back playing Shield cricket.
"I've bowled 30-plus overs in the last two days," Richardson said.
"It's something I haven't experienced in a little while, so I'm expecting to be a bit sore tomorrow.
"But I'm still getting through, and if required to bowl tomorrow, I'm ready to go."
WA were in need of a hero after entering day two at 5-64, and their hearts would have sunk further after Cameron Bancroft (20 off 38) chopped on Webster from the first ball of the morning.
Stanlake (4-35) was able to clean up most of WA's tail, but the last-wicket stand between Rocchiccioli and Couch ensured the first-innings deficit was reduced to 84 runs.
