Benzema scored twice inside of the first seven minutes in an action-packed start to the game, but the hosts pulled one back soon after with a goal from Oihan Sancet to halve the lead.
Benzema broke the 400-goal mark in his professional career with a stunning strike from just outside the box in the fourth minute before adding another just three minutes later when an Athletic defender missed a clearance.
Ancelotti said of Benzema after the match: "What a forward, what a captain, a great player, a great man, a very strong personality, he's very humble, he's a top, top person."
Real, who were missing eight players due to a COVID-19 outbreak in the club as well as suspended midfielder Casemiro, stretched their unbeaten run to 15 matches in all competitions as they reached the mid-season break.
"It's an important win against a difficult team at a fantastic stadium, they're three points that are worth a lot in the table, but now we'll rest and recover those who weren't here today," Ancelotti said.
"Today we've had two or three players who've tested negative, they haven't had symptoms, so I think when we come back, they should all be ready."
Despite the furious start, neither team scored again in a highly entertaining first half at San Mames, with Carlo Ancelotti's side taking a 2-1 lead into the break.
The end-to-end play continued in the second half, but Bilbao couldn't find a way back into the match and Real Madrid extended their lead in LaLiga to eight over second-placed Sevilla, who have a game in hand.
Benzema was applauded by the Bilbao supporters when he was subbed off in a rare gesture of goodwill between rival teams.
"I like this kind of football, at a great stadium like this, it makes me happy," said Benzema after the match. "But the victory makes me even happier and more proud, we know they're a difficult team, it's good to win here.
"It was a good game for us against a very good opponent. We conceded a goal but we scored two and it's three points for us."
He made his second consecutive start after being out from the starting eleven for three months. The Belgium forward quickly found his stride to lead most of Real's attacking moves.
Ancelotti said: "Yes, Hazard fits the team. Personally he played well in this game. We had a doubt about the physical side but he did well, he fought, he's back, yes, I agree."