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Dusan Tadic whipped a number of dangerous balls into the box throughout the first half and one such cross in the seventh minute found the head of Graziano Pelle, but the big Italian nodded wide of goal.
After half-an-hour, Pelle got another sight of goal following more good approach play from Tadic, but Jonas Olsson threw himself in the way of the shot.
Ben Foster almost handed the home side the lead when he came charging out of his area and got nowhere near the ball, but Ryan Bertrand opted to cross the ball for Pelle rather than shoot from a tight angle, and the ball sailed harmlessly to safety.
The England international goalkeeper made amends shortly afterwards with a good low save from Tadic's strike to keep his side level at the break.
Southampton introduced former Albion striker Shane Long at the break and record-signing Brown Ideye appeared five minutes later for the visitors as both sides looked for more cutting edge in the final third.
The lively Tadic tested Foster with a powerful strike before Ideye headed into the side-netting at the far post as the game began to show signs of life.
With 20 minutes remianing, Tadic fired yet another effort just wide of the post before James Ward-Prowse's free-kick rattled the frame of the goal.
Ideye, who cost the Baggies £10m, looked a much more likely source of a goal than Victor Anichebe did in the first half and was only denied an opener by the quick-thinking Forster, who rushed out to close down the angle and push away the striker's effort.
With just a few minutes remaining, Ideye looked like the favourite to latch onto a Dorrans through ball, but Forster was out quickly to thwart the Nigerian once more and ensure that he kept a clean sheet on his home debut.