The Hammers fought back from 3-0 down to earn a point.
Demba Ba's double - on his first start for the Londoners - and a header from Carlton Cole in between ensured the match finished 3-3.
"I think we have what you need to stay up," Grant said.
"If you look at other teams, if they had the problems we have had this year, I'm sure after the first half, the spirit would be very low.
"But this team, despite what we have had this season, have shown a lot of character and spirit and I don't think another team in this position could do that."
In the absence of Keane (calf) and Obinna (ankle), Grant was delighted to see Ba shine up front.
"It is the first game that he started and it is not easy," Grant said of the former Hoffenheim forward.
"In the first half you could see that he was not used to English football, but he adjusted very well and scored two goals. He is a good striker."
Asked what he had said to his players during the interval, Grant joked: "It is better that you don't know!" before adding: "We just spoke about how to change the game and the way we were playing because we had started the game not organised at all.
"We gave them a lot of possession and didn't defend well. If you don't defend, you see what happens, but we spoke about this."