England worldwide Hart was tested by Shaun Whalley and Mat Sadler, while the Premier League side mustered just four attempts at goal - only two on target - at the New Meadow. After Cheikh Kouyate failed to get a telling touch to a risky cross by Cullen in the ninth minute, any whiff of a first-half breakthrough by the away team vanished.
And Shrewsbury-born Hart admitted he needed help in the first half.
Cullen was taken off to be cleaned up and to change his bloodied kit, but did return to the field of play.
Hart, though, did not make a meaningful save in the second half, despite Town's efforts, and West Ham were let off the hook with four minutes left. "It doesn't feel like the Christmas period unless you're playing loads of games and I love it", the defender said. "We showed that we can go toe to toe with them and even outperform them, so I don't see why we can't go to their place and do the same thing".
Hart earned praise from boss David Moyes after being recalled following four games on the bench, while the England global has not played in the Premier League since November.
"I got a good rest on Saturday night expecting to have sore hands today but to be honest with you it was probably one of the easiest games I've played in all season".
"It's going to be a big test going there on a nice pitch when they are knocking the ball around, but we will look forward to that", Henderson added. Meanwhile, Moyes has already tasted FA Cup defeat to Shrewsbury once in his managerial career, when in 2003 his nascent Everton side were beaten by the Shrews in the third round.
"I should be surprised but I wasn't. Maybe a few of the boys were exhausted, we have had a lot of games recently".
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