On a day of chaos when 18 wickets fell, WA went to stumps at 6-79 in their second innings, but still sitting pretty given they have an overall lead of 331 with two days to play.
In reply to WA's first-innings total of 5-406 declared, South Australia were skittled for 154 in 37.1 overs, with Richardson (3-29) leaving the two-time defending champions reeling at 3-24.
Richardson had Mackenzie Harvey (2) caught at slip before finding the edge of Nathan McSweeney (1) two overs later.
The right-arm paceman was on a hat-trick after Hilton Cartwright took a juggling catch at slip to dismiss Jason Sangha for a first-ball duck.
Richardson's hat-trick ball was a beauty - pitching just in line with the stumps and staying low to crash into the pads of Jake Lehmann.
The entire WA team erupted in a huge appeal given the ball looked to be on track to take out the off stump, but it was turned down by the umpire.
Spinners Ashton Agar (2-35) and Corey Rocchiccioli (2-10) chipped in with two wickets apiece, alongside paceman Liam Haskett (2-28) as South Australia folded meekly.
Jake Weatherald (40) top-scored for the home side before being bowled by Agar, while Ben Manenti finished unbeaten on 38.
WA's second innings never got going, with Brendan Doggett (3-19) and Liam Scott (2-17) doing the bulk of the damage as the visitors slumped from 1-35 to 6-74 late in the day.
It was in stark contrast to day one when only three wickets fell.
WA opener Cameron Bancroft walked to the crease on Thursday unbeaten on 156 and with his side well placed at 3-304.
The former Test opener made his way to 186 before he was brilliantly caught on the boundary by Doggett, who had to chuck the ball up as he stumbled over the rope, before stepping back in to complete the catch.
Corey Wasley made a brisk unbeaten 60 off 57 balls to help build WA's total before they declared.
South Australia's huge collapse meant Bancroft got to bat again on Thursday - and he was out for 23 when he edged Scott through to the wicketkeeper.Â
