Richards struck five minutes into the second half to make it 2-2 after Argyle had recovered from conceding a second-minute Antony Kay goal to grab an interval lead.
Some fans were still taking their seats when Barnsley, trying to shake off a run of three Championship defeats, burst forward down the left and Kay spotted Luke McCormick off his line and unleashed an unstoppable shot from 30 yards to claim his first of the season.
Plymouth, with three straight wins behind them, were unfazed and earned a string of corners that saw them come close to an equaliser.
Tony Capaldi curled in a three dangerous flag-kicks and Barnsley were fortunate from one when Barry Hayles squeezed in a header that hit the post before Nick Colgan smothered a second effort.
And when Capaldi whipped in a testing free-kick from an angle Marcel Seip saw his header go over the top.
Midway through the first half strong running by Hayles almost paid off when he beat Colgan to the ball and pulled it across the face of an open goal, but nobody had got forward to finish off.
The equaliser came after 29 minutes when Sylvan Ebanks-Blake chased a long clearance from McCormick, sped past Neil Austin and crashed in a shot beyond Colgan.
Two minutes later only a tremendous save stopped the visitors going in front, when Colgan turned round a fierce low shot from David Norris.
Then, after 34 minutes, Plymouth went in front when Hayles chested down a cross from Norris and volleyed past Colgan from just inside the box for his fifth goal of the season.
A minute later Richards cut into the area from the left and tried to find a gap but saw his low shot cannon off the post, before Michael McIndoe and Paul Hayes had long-range shots saved.
Five minutes into the second half Barnsley were back level. McIndoe found Kay, whose low ball was parried by the keeper but Richards was first to react and crashed in a shot from 12 yards out.