Cardiff and Tottenham scored stoppage-time goals in the last two games at The Hawthorns to claim 3-3 draws, with Pepe Mel's side dropping to just three points above the drop zone.
Premier League fixtures at the home of the Midlands outfit have been dramatic affairs in recent weeks, with both Cardiff City and Tottenham scoring stoppage-time goals to secure a point in six-goal thrillers.
The two sides played out a 3-3 draw in December themselves, as Saido Berahino's 69th-minute goal ensured West Brom took a point from a game that hit the headlines for Nicolas Anelka's controversial 'quenelle' gesture in a goal celebration.
However, Allardyce hopes his players can steer clear of a repeat as West Ham goes in search of a victory that would see the club hit the 40-point mark and all but secure safety.
"[It will be] really tough," he told reporters. "They've had a resurgence and at their home ground they've been so close to so many victories and only slipped up in the dying seconds of two particularly thrilling 3-3 draws.
"We don't want to be in that position where we're trying to fight our way back from behind on Saturday. We want to be keeping a level playing field for as long as possible and our goal would be to do what we did at Sunderland, which was score the first goal and then go from there. The pressure in that game will be pretty intense because we all know what's at stake. We have to handle it better than West Brom."
West Brom remains embroiled in the fight against relegation, sitting three points clear of the bottom three with a game in hand, and defender Jonas Olsson knows a victory would ease the pressure significantly.
"We need to get over that line sooner rather than later," he told the club's official website. "And why not on Saturday?
"We will focus totally on that and get ready for it. Of course it will be a different game to [the 3-1 loss at] Man City but we will speak about it and we will have time to prepare and make sure we're ready."
West Brom has won fewer matches than any other team in the Premier League this season, but welcomes a West Ham side which has lost six of its last eight.
Gareth McAuley (calf) could make a return to action on Saturday, but Liam Ridgewell (knee) will be unavailable for selection.
West Ham's only injury worry is Marco Borriello (calf), although Matt Taylor could miss the match for personal reasons.