Moyes' side will be thankful for Joe Hart after the England global goalkeeper continued to deny his former boyhood club, however, supporters took to Twitter to lament the performance of one of their other players.
The hosts had by far the better chances as their Premier League opponents struggled to get a foothold in the cup-tie.
League One promotion chasers Shrewsbury grew more and more into the contest, however, and had a couple of big chances to open the scoring.
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.
West Ham needed Shrewsbury-born Hart, who began his career with the Shropshire side, to save well from Mat Sadler and Shaun Whalley while Jon Nolan fired over late on. Hart evidently forgot to pack appropriate headwear so shortly after kick-off, in order to shield his eyes from the low-hanging sun, he had to don a cap thrown to him by one of the travelling fans. I don't think we did that at all.
"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". Other clubs have done it, and if we ever won the FA Cup that would be massive for us, and for me.
"We are going to be on the back foot now and again, but hopefully we can ride the storm, catch them on the break and see what happens".
"The atmosphere was sensational - I wish we had that every week", Canaries academy product Morris said. We were up on them and rattling them the whole game.