Wayne Rooney equalised after a mistake by Ricardo Carvalho but Ballack struck the winner from the penalty spot after 86 minutes following a handball by Michael Carrick.
And Grant added: "We did what we needed to do. We won the game and put them under pressure.
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Grants says Chelsea dominated the game
"This team has a lot of character and we played good football today."
Victory at Stamford Bridge would have effectively guaranteed United the title, but nothing will be decided now until at least Monday, 5 May, when Chelsea travel to face Newcastle two days after United host West Ham.
If United beat the Hammers, Chelsea will need at least a point at St James' Park to ensure the race goes to the final day, when the Blues play relegation-threatened Bolton at home.
Grant added: "We won against Arsenal at Stamford Bridge and now Manchester United but we need to make sure we win our last two games."
Skipper John Terry felt the result against United showed that Chelsea are good enough to win the title.
"We want to win this Premier League and take the title from Manchester United," said the Blues skipper.
"I think we showed we can match anyone on our day."
After Ballack's opening goal, several Chelsea players held aloft a replica shirt with Pat Lampard RIP on the back, a reference to the mother of absent team-mate Frank, who died this week.
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And Grant said after the match: "I want to express my sympathy to Frank Lampard.
"We are thinking about him. It has been a tough week for him. I hope he feels good as soon as possible.
"You saw the players with the shirt after the goal. We are a big family at Chelsea."
Goalscorer Ballack said: "We have to say the victory is for Frank and his family at this difficult time."
