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Sunderland then threatened in the seventh minute when Jermain Defoe collected a long diagonal ball before going wide and crossing to Connor Wickham, whose volley from range was parried away by Adrian.
The Black Cats suffered an early blow when Brown landed awkwardly after appearing to be caught by Sakho, and despite getting up, he was not able to continue on the pitch, having to come off for Santiago Vergini.
Defoe was next to threaten as he sent a volley towards goal on the quarter-hour mark, but the ex-Hammer's effort flew wide of the post.
Alex Song was next to try his luck as he latched onto a headed clearance to smash a first-time volley from 30 yards, which required a diving save from Pantilimon to stop it from opening the scoring.
Sunderland looked the most likely to score at the start of the second half, with Wickham shooting over from long range before Jack Rodwell failed to convert from a corner.
Both teams failed to threaten again until the 77th minute when Patrick van Aanholt's long-range effort was heading towards the top-left corner, but Adrian tipped the ball wide.
Sakho finally broke the deadlock in the 88th minute, driving the ball into the bottom-left corner to give West Ham the three points.