United's 3-1 win on Saturday put them back on top of the Premier League by two points but Chelsea will go back to the summit by a single point if they can overcome Stoke at Stamford Bridge on Sunday.
Chelsea's players have spent much of the week getting over their 2-1 reverse by Tottenham last weekend but they have the best home record in the top flight with only five points dropped from 17 games.
Carlo Ancelotti's side have won all three matches since the Manchester City reverse with 12 goals scored and two conceded and Drogba knows how important the home fans can be.
"You never know what can happen, games are difficult now," said Drogba. "Some teams are trying to save lives but some teams are playing for little and can enjoy themselves, and sometimes then you play better.
"It is going to be difficult, but it is up to us. We need the fans to be really behind us, to be even better than against Bolton in the last game.
"Sunday is the kind of game that we are going to have until the end of the season so we need them to be patient and to really shout and sing and be supportive, that is the only way to win it.
"I remember the games when we were champions, like against Charlton the game after we won it in 2005, and against Manchester United the next year. The atmosphere was amazing. These are the kinds of atmosphere from the Chelsea supporters when it matters, and now it matters. It is important for them to be there and help us to achieve it."
Following the setback against Spurs, Drogba believes the squad have done all they can on the training pitch to get back on track.
"As always we work hard in the week," he said. "Sometimes it is not only because you work hard that you win at the end of the week but we will do everything in our powers to arrive there with fresh legs.
"The motivation has to be high because there are only three league games to go and three wins and we are there."
In-form winger Florent Malouda is adamant there is no panic within the dressing room as they approach their must-win game with City.
"The manager was really upset after the game at Tottenham and we as players were disappointed with that performance because when you are thinking about winning the league, that is not the kind of performance that you are wanting to do," said Malouda.
"You have to analyse what went wrong but we had two days off and we had to clean out our heads and then come back with a different attitude.
"The manager is always the same, he is staying cool and in control, and you think in a positive way and remind everybody that we are still one point in front. It is in our hands. Manchester United would like to be in our position and we must never forget that."
Meanwhile Ghana midfielder Michael Essien, 27, has admitted he faces a race to be fit in time for the World Cup finals after ruling himself out of Chelsea's season.
The Ghanaian midfielder, missing since December, failed to play in the majority of the African Nations Cup in January after sustaining a meniscus tear in training.
"Unfortunately I don't think I will be making a repeat of last season and may not be returning for the last few games and the FA Cup Final," he admitted. "As far as Chelsea goes I think my season is finished.
"I'm not too sure about the World Cup just yet. I will just have to wait and see what happens and take it day by day.
"Unfortunately you can't control when injuries will happen, but knowing the World Cup is on the horizon is certainly an incentive to keep working. But, I also have to keep an eye on the bigger picture.
"The most important thing now is to get myself back to full fitness. If I can make it in time for the World Cup then great, but if not I still have my whole career ahead of me and that is more important than anything.
"The last thing I want to do is rush back too soon and cause further problems down the line."