Eto'o put Chelsea ahead in the 56th minute and Hazard added a penalty four minutes later after Younes Kaboul had been sent off for a foul on Eto'o, with Demba Ba adding a late brace.
Eto'o's opener was gifted to him by Jan Vertonghen and both of Ba's goals also stemmed from errors, as Spurs' 23-year wait for victory at Stamford Bridge went on.
Chelsea now lead Liverpool and Arsenal by seven points, but both teams have a game in hand, while fourth-place Manchester City, who are nine points off the pace, have three fixtures to fulfil.
After Fernando Torres was injured in the warm-up, Eto'o came into the starting XI and he gave Spurs an early scare when he raced through and rounded Hugo Lloris, only for an offside flag to stop him in his tracks.
Hazard also took the ball past the Spurs goalkeeper moments later, but could only shoot wide, while at the other end, Nabil Bentaleb dragged wide from a promising position and Petr Cech saved from Sandro and Kaboul.
Spurs were very much in the game, but Vertonghen's blind pass 11 minutes into the second period enabled Eto'o to run through and score.
The Cameroon star celebrated with relish, referencing manager Jose Mourinho's recent comments about his age by staggering towards the corner flag like an old man.
Kaboul's innocuous challenge on Eto'o saw him dismissed and enabled Hazard to add a second, before Ba struck twice in the dying stages.
His first, in the 88th minute, saw him pounce after Sandro was unable to cut out a low cross from Oscar and a minute later a calamitous back header from Kyle Walker allowed the Senegalese striker to complete his brace.
Goals from Phil Jones, Wayne Rooney and substitute Danny Welbeck gave United back-to-back league wins for the first time since December, following their 2-0 success at Crystal Palace last month.
Victory took the defending champions to within nine points of fourth-place Manchester City, although they have played two more games.
"I thought our forward play was very good," United manager David Moyes told BT Sport.
"I thought both Robin (van Persie) and Wayne were excellent today, and Juan Mata and Adnan (Januzaj). But it also needed a good defensive performance as well, so all round we were pleased."
United's last game at The Hawthorns had been Alex Ferguson's final match as manager and his successor was looking for a reaction to his side's shock 2-0 loss to Olympiakos in the Champions League.
After Rafael da Silva had seen a header tipped onto the crossbar by West Brom goalkeeper Ben Foster, United took the lead in the 34th minute when Jones converted a free-kick from Van Persie with a glancing header.
Rooney headed home from Rafael's cross to make it 2-0 in the 65th minute and then fed Welbeck to run through and curl a shot past Foster with eight minutes remaining.
Steven Caulker scored twice for Cardiff, either side of a Lewis Holtby equaliser, before Sascha Riether's own goal sealed a victory that left Fulham four points adrift of safety at the foot of the table.
"We have got nine games and 27 points to play for, which is a lot, but in our situation we have to win our home games like Cardiff have done today," Fulham manager Felix Magath said.
Palace are only two points above the bottom three after losing 1-0 at home to Southampton, who prevailed thanks to a breakaway strike from Jay Rodriguez in the 37th minute.
The Republic of Ireland international converted a 73rd-minute penalty to cancel out Bradley Johnson's header, but was then sent off for an ugly foul on Norwich midfielder Alexander Tettey.