The Serbia international insisted that despite the busy schedule, winning the FA Cup is important to the players and fans, especially given an eight year spell since they last won the trophy.
Speaking after the match the centre-half told reporters: "We did not win the trophy for the last eight years and it's a trophy we are looking to win.
"We showed that in how we celebrated the goal, and I am hoping at home it will be a different game."
United conceded twice, both set up by debutant Joe Cole and finished by James Collins, but the Hammers were unable to see out the FA Cup third round tie, with Robin Van Persie's well crafted equaliser forcing a replay.
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"They scored from the second ball which we should look at again, and we are definitely a bit disappointed. It is something to talk about.
"We have the height in the team, we should deal better with that, but most important is that we managed to stay in the game."
Vidic was also quick to praise Robin Van Persie, whose late equaliser ensured that United remain in the competition.
Speaking to MUTV, he said: "His touch is magnificent and his finishing is top class. It is movement from the book, touch from the book, goal from the book.
"It is something they can show to kids, how to move and finish in the last third. He was magnificent for us, we just hope he is going to be in the same form."
The replay will take place at Old Trafford on January 16, with the eventual winners facing either Fulham or Blackpool.