Eto’o’s treble, his first in English football, gave manager Jose Mourinho his 100th Premier League victory, while United, who replied through Javier Hernandez, slumped to their seventh loss of the campaign and saw captain Nemanja Vidic sent off in stoppage time.
As well as the gulf that separates them from leaders Arsenal, United are also six points below the Champions League places, further jeopardising their chances of securing the consolation prize of a top-four finish.
With hopes of success via the league fading further, Wednesday’s League Cup semi-final second leg against Sunderland, when United will look to overturn a 2-1 deficit, now assumes even greater importance.
It was a fifth successive league win for Chelsea, meanwhile, who returned to within two points of Arsenal and a point of second-place Manchester City.
With Wayne Rooney and Robin van Persie both still injured, United fielded Adnan Januzaj in support of Danny Welbeck, but they quickly found a route through the Chelsea defence on a brisk afternoon at Stamford Bridge.
A neat one-two between Ashley Young and Welbeck presented Young with a clear sight of goal in the second minute, but his effort was parried by Petr Cech.
It was an enterprising start, and Chelsea found themselves on the back foot until Eto’o broke the deadlock in the 17th minute with a bolt from the blue.
Gathering the ball wide on the right, the Cameroon star tricked his way inside Phil Jones and then dispatched a left-foot shot that flicked off Michael Carrick’s toes and looped over David de Gea.
With David Luiz providing a muscular presence in front of the Chelsea back four, occasionally to the consternation of the United bench, the hosts began to assert greater control, but the visitors remained a threat.
Patrice Evra fired into the side-netting after blocking an attempted pass from Oscar, while Welbeck could only shovel a shot straight at Cech from Januzaj’s driven left-wing cross.
It was to prove a costly missed opportunity.
After Oscar had sent an acrobatic volley wide, Eto’o doubled Chelsea’s lead on the stroke of half-time when he stole in behind a static defence to clip home a cross from Gary Cahill after United failed to clear a corner.
Moyes sent United out early for the second half, but within four minutes the game was over and once again, the visitors’ defensive shortcomings were laid bare.
Cahill easily lost his marker to meet a right-wing corner with a downward header and although De Gea blocked, Eto’o was on hand to hook the ball home.
Moyes introduced Chris Smalling and Hernandez in place of Evra and Young, but even when United did procure a chance, fortune deserted them.
With 20 minutes remaining, Cech miscontrolled a back-pass in a perilous position, but as Hernandez slid in, the Chelsea goalkeeper somehow contrived to hook the ball away from him.
United finally found the net with 12 minutes remaining when Hernandez touched in Jones’s low centre, but by then the damage was done.
The visitors’ misery was completed in injury time when Vidic was shown a straight red card for chopping down Eden Hazard, while Rafael da Silva was fortunate to avoid the same fate for a rash two-footed lunge on Cahill.