Two first-half goals and one soon after the restart by Eto'o took Chelsea up to third place with 49 points, two behind leaders Arsenal, and United remained in seventh place, 14 off the summit.
United defender Nemanja Vidic was sent off in stoppage time to complete a miserable day for David Moyes's team.
United, and young attacker Adnan Januzaj in particular, started brightly but were undone by two poor pieces of defending that allowed Cameroonian Eto'o to score his first goals against United since the 2009 Champions League final when he was playing for Barcelona.
Eto'o opened the scoring after 17 minutes when allowed to shoot by Phil Jones, his deflected shot ballooning into the net, and he was left unmarked in the box just before the halftime whistle to double the advantage.
He completed his treble from close range when following up Gary Cahill's header to secure a fifth successive league victory for Jose Mourinho's side. Substitute Javier Hernandez pulled one goal back for United with 12 minutes remaining.
In the early match, Tottenham Hotspur secured a fifth successive away Premier League victory and moved up to fifth place after Emmanuel Adebayor scored twice in a 3-1 win at Swansea City.