Manuel Lanzini's first-half strike proved to be the difference between the two London rivals at Selhurst Park, although the hosts missed a penalty and struck the woodwork during the match.
The Hammers also had to hold out with 10 men for the final 15 minutes after Aaron Cresswell picked up two yellow cards in quick succession.
"We needed it as results haven't been the best of late, we had a bit of luck on our side. I thought we earned it and showed out there today we are a team all fighting for each other – this result will lift us," Reid told BBC Sport.
"It was nice that the formation paid off, and I thought we dealt with them as best we could and got forward well as well. We looked 100% as a team and were working hard for each other. We stay humble and keep working hard.
"I thought [Cresswell] was a little bit harshly done by, but the referee has made a decision and we had to dig in, which we did, and got a fantastic result."
The victory saw West Ham climb out of the relegation zone and up to 15th place in the Premier League table.