Victory left Jose Mourinho's Chelsea within two points of leaders Arsenal but this result meant title-holders United are 14 points adrift of the Gunners with 16 league games of the season remaining.
United's seventh league loss this term also turned the spotlight back on David Moyes, the successor chosen by Alex Ferguson, British football's most successful manager, to replace him at Old Trafford following his retirement at the end of last season.
The 32-year-old veteran Cameroon striker made it 2-0 on the stroke of half time when, following Gary Cahill's centre, he smashed the ball home from six yards out.
United, without injured strikers Wayne Rooney and Robin van Persie, found themselves 3-0 behind four minutes after the interval when more lax defending from a set-piece allowed Cahill a free header from a few yards.
His effort was just about blocked by de Gea but, barely a yard away, Eto'o reacted first and made no mistake with the simple chance.
In the process Eto'o become the first Chelsea player to score a hat-trick against United since Seamus O'Connell back in 1954.