Grant claimed the Hammers proved their limp performance against Newcastle was a one-off after watching them book a place in the quarter-finals.
Saturday saw West Ham throw away an early lead that would have seen them haul themselves off the foot of the Premier League.
The defeat piled the pressure on manager Grant and Stoke looked to have cranked it up even further on Wednesday evening when they led 1-0 with four minutes remaining at Upton Park.
But a late equaliser in normal time and two further goals turned the tie around.
Asked about the owners' unhappiness with Saturday's second-half performance, Grant said: "And you think that I was happy with the second half?
"I must say that the second half Saturday, it was not good.
"But if you see all our games - the last six, seven games - it was the exception.
"You want to show that it's the exception."
West Ham certainly did that on Wednesday night and Grant added: "We didn't start the game well but the reaction in the second half was great.
"We played well, we dominated the game, and we scored three goals."
The Hammers did not start well, Kenwyne Jones giving the visitors a sixth-minute lead which they came close to doubling before the home side turned the tide.
Grant said: "Even, the first half, we were playing well. We had two good chances the last 10 minutes of the first half.
"I thought if we continued to play like this, we would win the game."
Describing his half-time teamtalk, the Israeli said: "We spoke about the things that were not so good.
"We spoke about mentality and character, changing a little bit of the tactics.
"I'm very delighted because they responded."
He added: "I think one of the things that we wanted to improve in the team is their character.
"In the last weeks, we showed it.
"But Saturday, we didn't show it, so we wanted to show that Saturday was the exception for us."
Grant's substitutions turned the game, with Victor Obinna, Mark Noble and Valon Behrami all making a huge impact.
Captain Scott Parker got the equaliser but Noble set up the second and third goals, the last of which was scored by Obinna.
"When substitutes succeed, I am happy," said Grant, who hailed Noble's mazy run which set up the second goal as "like Maradona".
Of Parker, who once again led by example tonight, Grant said: "I've spoken so much about Scott Parker.
"I don't think any of us can say a negative word about him.
"He's one of the players that you're happy to have in your team."
Stoke boss Tony Pulis was philosophical about his side's third successive defeat.
"We're disappointed. We thought we played well first half," he said.
"I thought we had more opportunities. Our final pass was poor but we needed that second goal.
"West Ham threw everything at us in that second half, as you'd expect."
Stoke began to struggle once they brought off Jones and Jermaine Pennant, who set up their opening goal.
"The two substitutions with Kenwyne and Jermaine, who were under the weather a little bit, didn't help us," Pulis said.
"I thought Kenwyne was very good when he was on the pitch; he was a threat.
"But fair play to West Ham; you've got to give them credit.
"Obinna came on, young Noble, and they changed the game, really."
Asked if West Ham's comeback came as a surprise, he added: "It's cup football; they've got nothing to lose.
"We worked very, very hard for a good 75 minutes today and we tired a little bit at the end.
"But I've got to give them credit."