Since losing at Mallorca, Los Blancos have won four and drawn one of five games. PHOTO: AFP
Real were three up inside 29 minutes at Ipurua as Sergio Ramos scored a penalty between two Benzema strikes, the second also a spot-kick after Ramos delegated to his teammate. Fede Valverde added a fourth in the second half.
Benzema's double takes him to 157 goals for Real Madrid, above Ferenc Puskas into sixth in the club's all-time list, but another excellent display from the Frenchman was trumped by a creative masterclass from Hazard.
Madrid move top of La Liga, above Real Sociedad and Barcelona, who could return to the summit if they beat Celta Vigo at home later on Saturday.
But while Barca have lost their way after defeat by Levante and a goalless draw against Slavia Prague in the Champions League, Real will head into the international break as the league's form side.
Benzema stars as Madrid put five past limp Leganes
Since losing at Mallorca, they have won four and drawn one of five games, while scoring five against Leganes, six against Galatasaray and now four against Eibar. All five have yielded clean sheets, with the team now passing 533 minutes without conceding.
Their scoring problems have frustrated this season too, not least in a home stalemate against Real Betis earlier this month, but Hazard, and his burgeoning partnership with Benzema, could well provide the answer Madrid have been looking for since Cristiano Ronaldo left almost 18 months ago.
Hazard, checked by an early thigh injury, has taken a while to get going in Spain but this was a vintage performance from the 28-year-old to suggest he is finally finding peak form and fitness.
He was unplayable for the first half an hour, a series of mesmerising runs down the left leaving Eibar's Cote and Takashi Inui powerless to keep up.
His instinctive connection with Benzema came to the fore too. Benzema had already hit the post and he should have scored from a delicious Hazard cross before opening the scoring in the 17th minute.
Benzema latched onto Hazard's pass and after Valverde and Luka Modric missed, the ball spilled back to the Frenchman, who found the corner from an acute angle.
Hazard pulls strings as Real Madrid edge Galatasaray
Three minutes later it was two, Hazard the creator again, his feet too quick for Pablo De Blasis, who caught his opponent's right leg. Ramos slid the penalty home.
Hazard wanted a spot-kick when he was caught by the onrushing Eibar goalkeeper Marko Dmitrovic but another one would come soon. First, there was a brilliant nutmeg from Hazard, a 'rabona' cross and too many glides past defenders to count.
Ramos passed on the second Madrid penalty, awarded for a foul on Lucas Vazquez this time, to allow Benzema to make it three inside half an hour.
The second period was a procession and Hazard was taken off with 18 minutes left, his work already done. By then the dynamic Valverde had added a fourth, cracking in Modric's pass for his first Real goal in La Liga.
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