NEW YORK—Novak Djokovic heard what Andy Roddick said about him and didn’t like it one bit.

Still, as much motivation as Djokovic might have had, and as well as he played in their U.S. Open quarterfinal Thursday night, Roddick’s own uncharacteristic serving miscues had a lot to do with the 2003 champion’s 6-2, 6-3, 3-6, 7-6 (5) loss.

Two points from forcing a fifth set at 5-4 in the fourth, Roddick double-faulted twice in a row and was broken for the fifth time—twice more than he lost serve in his first four matches combined.

In Djokovic’s prior match, a five...

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