The London club hung on for a point at St Mary's Stadium despite seeing goalkeeper Adrian sent off for handling the ball outside the area.
Jussi Jaaskelainen is likely to fill in for the suspended Spaniard, although Allardyce has already confirmed West Ham's intention to appeal the ban.
Andy Carroll suffered a knee injury shortly after Adrian's dismissal, but continued to hobble around the pitch as the visitors had used all three substitutions.
And, with Cheikhou Kouyate limping off and Kevin Nolan (foot), Mark Noble (illness), James Collins and Winston Reid (both hamstring) all doubts, Allardyce is wary of his fatigued side coming unstuck at The Hawthorns.
"We've still managed to secure a point, but my fear is unfortunately that the extra effort we have put in with nine men will maybe take its toll when we play Saturday in the FA Cup at West Brom," Allardyce told West Ham's official website.
"Andy is out and there were some real tired legs out there and I saw Cheikhou limping off at the end after another terrific performance.
"I can only hope that Kevin and Mark are fit. I think it will come just too soon for James Collins and Reidy."
By contrast, Jonas Olsson (Achilles) and Victor Anichebe (groin) are West Brom's only injury worries as the hosts look to build momentum following their 2-0 Premier League win over Swansea City.
West Brom reached the semi-finals of the FA Cup back in 2008, and head coach Tony Pulis feels the Midlands club can afford to be confident on the back of their midweek success in the league.
"They [West Ham] are a top side having a very impressive season," Pulis told the West Brom's official website.
"We are a couple of games from Wembley with the semi-finals staged there.
"If we can get through this challenge and get a favourable draw...who knows what might be ahead? It is the excitement of the Cup and I know this club has a great history in the competition.
"I was brought up on the FA Cup and love the competition as well. I'm really looking forward to a top occasion on Saturday – I hope our fans are too.
"The players should be in confident mood after the performance against Swansea and I hope they and the supporters can pick it up where they left off on Wednesday."
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