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Cleveland extended their lead as they took advantage of a mistake by Patrick Robinson, who jumped into the neutral zone on a missed field goal, and Cundiff fired a kick through the posts from 32 yards.
However, just before the break, Brees drove his side down the field and he connected with Jimmy Graham in the endzone for the nine-yard score.
After the break, New Orleans pulled themselves ahead as Brees responded to the adversity he was facing from Cleveland's pass-rush by throwing to Graham in the flat for his second touchdown of the game.
Cleveland were not deterred after going behind and they put Johnny Manziel into the game at quarterback. His presence kickstarted his side's offense and West scored with a smart cutback run for a nine-yard touchdown.
Back came the visitors as Brees and Graham combined to put Sean Payton's men in the redzone, and Mark Ingram showed his ability as a power runner as he forced his way into the endzone from one-yard out.
There was to be a twist in the tale as Hoyer engineered a strong drive to move his side into the field-goal range, and Cundiff held his nerve in the last second to drill his kick between the posts and fire his side to their first win of the season.