United had begun brightly, but it was Chelsea who broke the deadlock with one of their first attacks on 17 minutes as Eto’o saw a shot deflect off Michael Carrick and over a helpless David De Gea.
De Gea could do little with Eto’o’s second which arrived just before half-time, as United failed to clear a corner and Gary Cahill crossed for the Cameroon star to fire home from six yards.
United looked to get back into the match early in the second half but the game was put well and truly out of their reach as Eto’o helped himself to his third just minutes after the re-start.
And it was again Cahill who played his part as he met Willian’s corner, De Gea could only parry and Eto’o was on hand to turn it home from close range.
There was time for United to get on the score-sheet as substitute Javier Hernandez converted what proved to be no more than a consolation for David Moyes’ side.
And to make matters even worse, United captain Nemanja Vidic was shown a straight red in injury time for diving in on Eden Hazard.
The three points sees Chelsea move back to within touching distance of leaders Arsenal and second placed Manchester City – although the win – Jose Mourinho’s 100th in the Premier League – does mean the top three are now six points clear.
United stay seventh, six points adrift of fourth place and 14 points off the top and the defence of their title surely now looks in tatters.
Both Chelsea and United are in transition, but it is clear who has more work to do after Mourinho’s men claimed a sixth win in seven Premier League games.
United’s attacking endeavours relied on 18-year-old Adnan Januzaj and that, coupled with the gaping defensive holes Chelsea exposed, increased the sense Moyes must recruit top-class reinforcements this month to chase a top-four spot.
Mourinho’s preferred Eto’o to Fernando Torres in a side showing two changes following the win at Hull.
The other was in defence, where Branislav Ivanovic returned earlier than expected from a knee injury so Cesar Azpilicueta shifted to left-back and Ashley Cole was dropped to the bench.
United, with Wayne Rooney and Robin van Persie again out injured, made three changes following the win over Swansea – Jonny Evans, Phil Jones and Ashley Young coming in – and began on the front foot.
But the Blues went ahead when Jones backed off Eto’o, whose shot looped up off Carrick’s out-stretched leg and into the top corner.
David Luiz, again playing in midfield, was booked for an over-zealous aerial challenge on Antonio Valencia which enraged the United bench.
If Luiz was the enforcer, Hazard’s influence as sorcerer was increasing as he found space to run at United.
Januzaj, who could yet team up with Hazard for Belgium, was United’s most creative force but no one was in the box to meet his dangerous cross.
Another Januzaj cross evaded John Terry and reached Danny Welbeck, who got a shot away under pressure from Azpilicueta, but Petr Cech saved smartly.
Chelsea doubled their advantage when the visitors failed to clear a corner.
Ramires found Cahill on the right side of the area and the defender centred around a static defence for Eto’o to sweep in.
It was a major blow prior to the interval, but the decisive one followed and featured more abject defending.
Cahill was allowed a free header from Willian’s corner and De Gea saved on the line, but Eto’o poked in the rebound.
United, who saw Evra depart injured, threw on Hernandez alongside Welbeck.
The Mexican pulled a goal back 12 minutes from time when he converted Jones’ shot after Welbeck’s pull-back.
But that was as good as it got for United, who saw Vidic sent off and Rafael booked in stoppage time as Chelsea triumphed.