The Hammers suffered what appeared to be a major setback early on, with the concussed Matt Taylor and Joe Cole, nursing a hamstring problem, having to come off in only the 10th and 11th minutes respectively. But it was their replacements, Jack Collison and Ricardo Vaz Te, who subsequently combined for what turned out to be a decisive goal in first-half stoppage time.
Allardyce said: "Our misfortune ended up as our good fortune today. Our misfortune was Matt Taylor going off with concussion and then straight after that Joe Cole with a hamstring injury. But Ricardo Vaz Te's ball to Jack Collison and (the performances of) those two players throughout the game today has helped us win this match."
He added: "We were a little desperate after only a few minutes when we saw two players coming off injured, but by the end of the game we had a well-deserved victory and it just shows the importance of the strength and depth of your squad.
"Two players come on and the team hasn't got any weaker - it stayed strong and continued to probe the Stoke defence, which finally paid off."
West Ham twice survived scares in stoppage time at the end of the match to hold on for the victory. First, Stoke substitute Charlie Adam cracked a volley against the bar, and the hosts then had appeals that Hammers defender Guy Demel had handled in the area waved away by referee Jonathan Moss.
Stoke manager Tony Pulis, whose side's defeat was only their second at home in the top-flight this term, was convinced the incident involving Demel should have resulted in a penalty, although he seemed more aggrieved about one in the build-up to the goal that featured West Ham striker Andy Carroll and Potters skipper Ryan Shawcross.
Vaz Te was able to slip in his pass to Collison after attempting to play a one-two with Carroll and having the ball come back to him via a coming-together between the England frontman and Shawcross.
Regarding that, Pulis said: "Their goal is really hard to take - Carroll actually pulls Ryan down and we are disappointed with that, and we could have had a penalty in the second half.
"We thought it was a definite handball. But like the offside rules, handball is a very ambiguous topic at the moment."