With West Ham having drawn two of their last four games in front of their own fans, those 12/5 (3.40) odds are well worth the investment on a typically tight and scrappy matching resulting in a stalemate for their seventh home game in eight versus West Brom.
Alan Pardew's team claimed a 1-1 draw with Arsenal on Sunday and now face a second game in 51 hours.
Whilst West Ham go into the game at the Montgomery Waters Meadow as favourites thanks to their Premier League status, Moyes doesn't believe that an extended run will be beneficial for his side.
Their Thursday game against Tottenham at Wembley was originally scheduled for the weekend just gone but club insiders said it was moved because of issues to do with the London Underground.
He said: "It's bad".
"We all know theres pressure on TV companies for ratings and everything else and games", said Albion boss Alan Pardew.
However, Moyes, who played for the Shrews between 1987 and 1990, has so far failed to right the ship completely, and has admitted that Premier League safety will always trump cup competitions in his view.
If the Scot wants to make changes then he can look to his bench, where options include Andre Ayew and Javier Hernandez.
"Of course I am. Theres some players who wont be able to cope with that", he said.
'We have got two days rest in Premier League professional football, putting players at risk, in my opinion, of injury. "Matty Phillips for example - I wouldn't take Matty Phillips off (against Arsenal) if I didn't have West Ham".
'If I could win the cup I would definitely try to win the cup, but I couldn't do it if I thought it would affect the Premier League'.