The Hammers suffered what appeared to be a major setback early on in the Barclays Premier League contest at the Britannia Stadium, 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, the latter playing a through-ball to the former, who turned and drilled a shot beyond Asmir Begovic.
Reflecting on a win that halted his team's run of five straight away league defeats, 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.
"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."