The home side dominated the fixture in the North-East but failure to score and Norwich City's dramatic 3-2 victory over Newcastle United has left Sam Allardyce's men in trouble at the bottom of the standings.
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West Brom began brightly at the Stadium of Light but it was Sunderland who created the first opportunity when Jermain Defoe met a cross from Younes Kaboul, but his flick was straight at goalkeeper Ben Foster.
Sunderland continued to pose a threat down the right flank, but Defoe's next chance came through his own work after turning sharply before firing wide of the far post.
West Brom were matching their hosts without threatening a goal and Sunderland almost capitalised shortly after the half-hour mark when Wahbi Khazri released Lee Cattermole, but the midfielder's effort was well smothered by Foster.
After the restart, it was one-way traffic as Sunderland looked for a crucial goal at the bottom of the standings, but they were unable to find the goal that their play deserved.
As the match entered the closing stages, West Brom continued to offer nothing in the final third and that gave Sunderland the platform to mount a final sustained assault on Foster's goal.
When they finally created a chance, it fell to Borini but the Italian flashed an effort marginally wide of the target, much to the frustration of the home side's supporters.
There was still time for Defoe to be handed one final opening for the home side but he was somehow denied by Foster from close range and Sunderland had to settle for a point that was not really enough given their battle to avoid relegation to the Championship.