After Shane Long had given the hosts the lead, strikes from Phillips and Robson-Kanu five minutes either side of the break proved decisive, as Tony Pulis' men go eighth in the top flight at the turn of the year.
Claude Puel made wholesale changes for The Saints, with six changes from the side that lost 4-1 to Tottenham Hostpur on Wednesday evening.
Out went the injured Cedric Soares, as Cuco Martina stepped in at right-back. Jose Fonte, Jay Rodriguez, James Ward-Prowse and Ryan Bertrand were rotated, with Shane Long, Dusan Tadic, Steven Davis, Pierre-Emile Hojbjerg, Sam McQueen and Maya Yoshida coming in. Nathan Redmond was suspended following his dismissal in midweek.
The visitors made just the one alteration, with Salomon Rondon earning a rest, allowing Hal Robson-Kanu a rare start up front.
Eventful end to opening period
After a largely scrappy and forgettable opening 40 minutes, the game suddenly exploded into life five minutes before the break. From a Dusan Tadic corner, Shane Long rose highest above the Baggies defence to score his first league goal of the season.
The hosts' lead however lasted less than two minutes. After Southampton failed to pick up Matt Phillips, the West Brom winger turned Virgil van Dijk and Maya Yoshida inside out, driving a low shot across Fraser Forster and into the Saints' net.
Baggies buoyant after break
The equalizer had visibly lifted a jaded looking visiting side and five minutes into the second period, West Brom took to the lead. Again Phillips was orchestrator, as his ball picked out the galloping Robson-Kanu. The Welshman looked up and smashed an unstoppable left-footed drive past Forster into the top corner, to stun St. Marys'.
Having conceded four against Spurs, Fraser Forster was picking the ball out of the net for the sixth time in a matter of days, with the natives restless. West Brom pushed forward for a third and came close through substitute James Morrison, before Rondon himself came narrowly close to converting Chris Brunt's teasing cross.
Southampton were almost gifted a leveller, when a Forster clearance clattered off Oriol Romeu, but the ball ricocheted wide. To compound their misery, Virgil van Dijk was dismissed late on for a second yellow card, to compound a miserable four day. Tony Pulis' men should have made it three with a four-on-two break in stoppage time, but West Brom eased to a comfortable 2-1 win.
The wins sees the Baggies leapfrog Southampton into eighth, with a home clash against Hull City on Monday. Puel's men will look to start the New Year on a more positive note, with a trip to Merseyside to face Everton and Ronald Koeman.