Mel, who will take charge when the Baggies host Everton in their next league game, will inherit a side with one win in 11 matches and their lacklustre display on the south coast underlines the job in hand.
West Brom's effort could not be questioned and emphasised why caretaker boss Keith Downing had suffered defeat just once in four matches ahead of kick-off, but his side's attacking intent was almost non-existent.
All too often they invited pressure, before changing tack after Lallana scored his sixth goal of the season to cap a typically busy display.
Downing's side opened the game in disjointed fashion and Steven Reid's wayward shot from 35 yards was all they had to show for their first-half endeavour.
Lallana a marked man
Ahead of kick-off, Adam Lallana had been dispossessed 48 times this campaign, more than any other player in the Premier League.
Southampton, despite their positivity, were unable to penetrate their opponents' defence, prompting wayward efforts from range by Lambert and Rodriguez.
4-1 rout of Hull in November,
Mauricio Pochettino's side have slipped from third to ninth, but they showed more willing in a second half in which their visitors looked happy to settle for a point.
Rodriguez seized upon a loose Youssouf Mulumbu pass but smashed a shot wide as the visiting defenders retreated, but Southampton were yet to forge a clear opening.
West Brom frustrated their hosts. often keeping 10 men behind the ball, but it was the introduction of Ramirez which led to their defence being breached.
The Uruguayan's lofted first-time pass from out wide found Lallana in space and the midfielder, who scored his first senior goal against West Brom in 2008, drilled home from 12 yards.
West Brom didn't do enough - Downing
The introduction of Baggies top scorer Saido Berahino offered the away side a much-needed edge late on, but it was Long who nearly equalised.
Liam Ridgewell's cut-back found the striker unmarked five yards from goal but Boruc, back after eight games out injured, produced a stunning save with his left leg.
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