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.
Eto'o gave Chelsea a 17th minute lead when his shot from just outside the box deflected off Michael Carrick's outstretched boot and past the despairing dive of United goalkeeper David de Gea.
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.
United pull one back
Javier Hernandez, on as a substitute for Ashley Young, pulled a goal back 12 minutes from time.
But that was as good as it got for United whose misery was compounded when captain Nemanja Vidic was sent off in stoppage time for a rash challenge on Eden Hazard.