Eden Hazard was a late withdrawal for Madrid through injury, while the absences of the likes of Marco Asensio, Brahim Diaz and Rodrygo Goes opened the door for Gareth Bale to be handed a surprise start.
Bale has been strongly linked with a move away from the Bernabeu this summer - he was handed a starting role in Vigo, though, with Vinicius Junior also chosen to support Benzema in the final third.
After a quiet opening to events, Madrid made the breakthrough in the 12th minute. It was Bale who provided the assist, with the Welshman doing brilliantly to create some space down the left before delivering into Benzema, who could not pass up the chance to find the back of the net from close range.
Celta had plenty of the ball in the period that followed, but the home side were finding it difficult to fashion opportunities, with Casemiro providing a strong shield for the likes of Sergio Ramos and Raphael Varane.
Bale continued to impress down the other end, meanwhile, with the 30-year-old's pace too much for Celta's back four to handle. That said, like the hosts, Madrid were also struggling to create clear openings.
Celta did have a half-chance in the 31st minute when Iago Aspas broke into a wide area before delivering towards Gabriel Fernandez, but Varane just managed to poke the low cross away at the vital moment.
Madrid's best chance of a second in the first period fell to Modric after the Croatian had worked some space in a dangerous position. However, Celta stopper Ruben Blanco was on hand to make a smart save, before the Spaniard also kept out Bale in the latter stages of the half.
The home side thought that they had levelled in the first minute of added time at the end of the first period when a mistake from Alvaro Odriozola allowed Aspas to feed Brais Mendez to fire past Thibaut Courtois. However, the resulting VAR check found that Aspas had been slightly offside, allowing Madrid to stay ahead.
Benzema had a brilliant chance to send Los Blancos two goals ahead in the early stages of the second period, but the centre-forward could only send Marcelo's low pass wide of the far post.
Courtois then denied Aspas with a smart save down the other end during a spell where both teams were enjoying positive moments in the final third of the field.
Madrid were incredibly reduced to 10 men in the 56th minute when Modric was handed a straight red card for a challenge on Denis Suarez. Initially the referee had only issued a free kick but after looking at the screen at the side of the pitch, ultimately decided to send the midfielder from the field.
Nestor Araujo almost levelled moments later when he found himself free inside the Madrid box, only for Courtois to pull off an excellent stop to keep Zidane's side ahead.
Madrid actually doubled their lead in the 61st minute when Kroos worked a yard of space outside the Celta box before finding the back of the net with a sensational strike via the crossbar. It was a thunderbolt of an effort from the Germany international, who had given his side some breathing space.
Aspas just missed the Madrid post with a curling effort in the 73rd minute as Celta desperately looked to halve the deficit entering the final 15 minutes of the contest.
The match was ended as a contest in the 80th minute, though, when Lucas found the back of the net with a smart finish following an excellent passing move.
That said, the home fans did have something to cheer in the final moments when substitute Iker Losada found the bottom corner with a brilliant finish.
Los Blancos have won on five of their last six visits to Celta in the league, and the victory was huge for Zidane's side considering the circumstances. Next up for Madrid is a home game with Real Valladolid next weekend, while Celta will attempt to put their first points on the board when they host Valencia.
CELTA VIGO (4-4-2): Blanco; Kevin, Araujo, Costas, Olaza; Beltran (Diop 83'), Lobotka (Sisto 75'), Suarez, Mendez; Aspas, Fernandez (Losada 88')
REAL MADRID (4-3-3): Courtois; Odriozola, Varane, Ramos, Marcelo; Modric, Casemiro, Kroos; Bale (Isco 75'), Benzema (Jovic 81'), Vinicius (Lucas 69')