The victory meant that they finished a point ahead of Manchester United, who beat Stoke City 4-0.
When Drogba headed the fifth, it carried Chelsea to 100 goals for the season, the first time any side has reached the mark in the English top flight since 1963.
If Chelsea had any nerves, they disappeared after six minutes.
A Drogba free-kick hit the wall, but as Frank Lampard returned the ball to the box, Florent Malouda, who was possibly offside, chested the ball down for Anelka to sweep in.
Wigan, fielding an unfamiliar three-at-the-back formation, began to dominate possession, although without ever really posing a threat, and the game was as good as ended as a contest after 32 minutes.
Lampard was hauled down in the box by Gary Caldwell, who was sent off, and stood up to rattle in his 22nd Premier League goal of the season.
In Image: Chelsea's players celebrate winning the Premier League title with the trophy after defeating Wigan Athletic at the end of their English Premier League soccer match at Stamford Bridge, London, Sunday May 9, 2010
Text: Agencies
Images: AP