Grant pits his wits against Potters boss Tony Pulis in the fourth round of the competition for the second straight year after managing Portsmouth to a 4-0 victory over the Staffordshire side last season.
More importantly, relegation-haunted Pompey took the confidence from that win into their next Premier League game against Wigan, who they thrashed by the same scoreline.
"It would be very nice to repeat that," said Grant, who is back under pressure at the rock-bottom Hammers after their five-match unbeaten run came to an end on Saturday.
"I remember the game well. It was three weeks after I arrived and it was a great game. After that, we also won a big game in the league and the football side improved."
That pattern has continued somewhat at Grant's new club, with West Ham following up last month's Carling Cup third-round win at Sunderland with their first league victory of the season against Tottenham.
Grant added: "The Carling Cup is important. We won at Sunderland in the last round and this gave us a big lift forward for the league. We played well against a difficult team and then won against Tottenham. We want to do the same again."
West Ham have already had a semblance of success against Stoke this term, ending their four-match losing start with a 1-1 draw at the Britannia Stadium three days before their Carling Cup win at Sunderland.
Grant said: "It will be a big test. Against Stoke, it is always a battle, a physical and mental battle.
"You always need to be awake, as it is a different style of football.
"We did well at Stoke in the league last month and I hope we reproduce that."
Saturday's defeat to Newcastle left West Ham rooted to the foot of the Premier League but the atmosphere in the camp has been upbeat since, according to Grant.
"I have been surprised by the mood," he told the club's official website.
"The players are disappointed, of course, but they are more determined to get back to the form that we have shown in the past month.
"We had been much better than we were at the start of the season and enjoyed a good unbeaten run.
"Unfortunately, we lost the game on Saturday. We cannot change these things but we can learn from it.
"We need to focus on the positive side. We have learnt from our mistakes before and come back strongly."
Grant is set to be without captain Matthew Upson and striker Frederic Piquionne on Wednesday night, with both players going off injured against Newcastle.
"The medical department are doing their best to get as many players ready as possible," said Grant, who will introduce one or two fringe men to his squad.
"We have a few players that are coming back from injury and we need to protect them. It is not easy.
"We want to use them. It is difficult when you play two times a week but we will do the best we can."