Cruising at 2-0 as the game entered first-half injury-time, the Pilgrims switched off and allowed Kevin Philips to net twice before the interval.
The former England man first capitalised on some dozy defending by the Argyle rearguard to take the ball round French goalkeeper Romain Larrieu and slide his shot home.
Then Larrieu made a mistake which ultimately cost the Pilgrims all three points, mis-kicking Tony Capaldi's back-pass straight to Phillips, who returned it first time form 40 yards over the goalkeeper's despairing dive.
Up until then, Argyle had commanded the game with some lovely flowing, creative football which the Baggies did well to match.
The home side took the lead midway through the half when David Norris drove through the centre of the Baggies defence and Barry Hayles drew goalkeeper Russell Hoult and slid the ball into the net.
They doubled their advantage in the 40th minute after Paul Robinson had fouled Sylvan Ebanks-Blake on the Argyle right. Capaldi slung over the perfect free-kick and French midfielder Lilian Nalis rose unmarked to head the ball in.
Man of the match Ebanks-Blake enjoyed a splendid start, nearly heading home after Norris and Marcel Seip had combined well on the right, and then teeing up Norris for a shot that Hoult was happy to see go wide.
After the break, the game was real end to end drama as both sides threw caution to the wind.
Larrieu redeemed himself with a save from John Hartson's fierce drive before Hoult almost committed a carbon-copy of the Frenchman's first-half error by kicking a clearance straight at Ebanks-Blake. However, Curtis Davies managed to clear the ball from the line.
Jonathan Greening set up Phillips for his hat-trick but Larrieu stopped that effort on the line and then saved Jason Koumas's long-range free-kick.
Ebanks-Blake's overhead kick found Hoult equal to the task, before Norris lifted a shot over the crossbar.
Larrieu continued to go from villain to hero after Phillips was offered the chance to complete his hat-trick by Hayles' misplaced pass. Larrieu, though, stuck out a long leg to thwart the striker.
Argyle substitute Cherno Samba, on for broken cheekbone victim Bojan Djordjic shot wide, before fellow sub Neil Clement was sent off for the Baggies in injury time for a second yellow card.