It is the fourth league title in 11 seasons, but only the fifth in Chelsea’s 110-year history.
Falcao was once one of the hottest properties in world football but a serious knee injury has curtailed his career.
On the final day of the season, Hull City were relegated while New Castle escaped narrowly.
The Ivory Coast striker, who first signed for Chelsea in 2004, has scored 104 goals in 253 league matches.
Sunderland’s American owner, Ellis Short, was in north London to see his team play out a goalless draw.
Chelsea are eight points clear of City going into their final match of the season.
Chelsea have 84 points with two games to play, City have 76 with one to play.
Frenchman Gomis also scored the winner when Swansea beat Arsenal 2-1 at home in November.
United will be guaranteed a top-four finish if Liverpool loses to newly crowned Chelsea on Sunday.
Chelsea have 80 points from 34 games with last season’s champions Manchester City second on 67.
Fernandinho volleyed in an 89th-minute winner as City survived Villa scare.
According to Football Association Fair Play delegates, Chelsea the worst-behaved team.
The win closed the gap on fourth-placed Manchester City to four points with six games remaining.
Defeat kept fourth-placed Manchester City nine points behind leaders Chelsea.
Liverpool manager Rodgers said that his club is not looking to sell the 20-year old youngster.