Benzema displayed exquisite control to get Madrid rolling in the 23rd minute. Spinning away from the goal, the French striker used his right foot to cushion a forward lob by Luka Modric. He then continued his turn and coolly slotted the ball in.
After he went close by hitting the crossbar, Benzema helped produce a second goal for the visitors in the 56th. Benzema dribbled past a marker and fired a shot that ricocheted off goalkeeper Sergio Alvarez, a post, and Celta defender Gustavo Cabral before finally going in.
Hugo Mallo pulled one back for Celta and Iago Aspas forced goalkeeper Thibaut Courtois to get low to parry his shot before Sergio Ramos put the result beyond doubt in the 83rd. Ramos scored from the spot by softly chipping his penalty down the middle after David Junca fouled Alvaro Odriozola.
Celta finished with 10 men after Cabral was shown a second yellow card in the 87th.
Dani Ceballos then added a fourth goal for Madrid from long range before Brais Mendez scored Celta's second.
Solari has coached Madrid to four wins since he was named caretaker coach following the firing of Julen Lopetegui. The club was looking for a permanent replacement, but Solari may have convinced Madrid that he should stay on.
Solari refused to talk about his future after the win, saying "the important thing is to do what is right for the club."
Madrid was also reticent.
"What I can say is that we are very happy (with Solari)," club spokesman Emilio Butragueno said.
Madrid remained in sixth place, but closed to within four points of leader Barcelona, which lost 4-3 to Real Betis.
Madrid was already without injured defenders Marcelo, Raphael Varane and Dani Carvajal. It lost three more players to injuries during the game at Bailados Stadium, including midfielder Casemiro, who hurt his right leg in a run-in with Celta's Maxi Gomez.
Solari also said that Gareth Bale played the second half with a swollen ankle.