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Mile Jedinak should have put Palace in front in the ninth minute when he latched on to Puncheon's cross, but the midfielder could only head wide to the right as the goal stood gaping.
Puncheon made amends, however, in the 16th minute when he found himself with space inside the box, allowing the winger to convert a left-footed shot into the top-right corner of the Hull goal.
Adrian Mariappa tried to double the hosts' lead with a right-footed shot from outside the box, but it was wasteful as the ball rose over the crossbar.
It took 30-plus minutes for Hull to register their first meaningful attempt on goal as Tom Huddlestone let fly from range, although it was blocked when things seemed dangerous for the Palace keeper.
Jedinak thought that he had scored a vital goal to increase his side's advantage as he slammed in a free kick, which Allan McGregor did well to turn over the bar.
Yannick Bolasie should have doubled Palace's lead just before half time but the winger spurned his chance by heading wide from only a few yards out from goal.
Jelavic then delivered a perfect ball for Jake Livermore, but the midfielder lashed his effort over the crossbar.
Long should have grabbed the visitors a late equaliser when he was found by Ahmed Elmohamady's ball, but the striker headed over the bar from only a few yards out.