Valverde's decisive strike came deep into added time, with the midfielder's shot taking a heavy deflection off Marcos Alonso before looping into the net. The late goal ensured Real Madrid claimed their first league win in three matches after a recent dip in form.
Earlier in the match, Aurelien Tchouameni had given the hosts the lead. The midfielder struck shortly after Vinicius Júnior had rattled the post, with Tchouameni's follow-up effort bouncing in off the woodwork.
Celta Vigo fought back to level the contest after a defensive lapse. Trent Alexander-Arnold was beaten on the wing by Williot Swedberg, whose cross set up Borja Iglesias to score the equaliser.
Real Madrid were led on the touchline by Alvaro Arbeloa and had to cope without several key players due to injury. Among the absentees were Kylian Mbappe, Jude Bellingham, Rodrygo and Eduardo Camavinga.
The victory moved Real Madrid to within one point of league leaders Barcelona. Barcelona can restore a four-point advantage when they face Athletic Club on Saturday.
"There's a long way to go because now, in every match, you're going to be playing against teams that have their own objectives. And every match is going to be a battle," said Arbeloa.
"Given how we came into today's match, with ten players out, if I'm not mistaken, it's a huge achievement.
"I think the players deserve a lot of credit for the match they played on a pitch like this; we all know how tough and difficult Balaidos is. Given how Celta had been playing, with the confidence they had been showing.
"So, of course, I'm very, very happy... above all, for the character, for the people who wanted to come, who wanted to push. Who, at a difficult time, wanted to take responsibility... and as I said yesterday, these are the moments when you see true Real Madrid players. And today I saw many of them."
Attention will now turn to Europe for Real Madrid, who are set to host Manchester City on Wednesday in the first leg of their UEFA Champions League last-16 tie.
Manchester City face uncertainty over Erling Haaland's availability as they prepare for their Champions League Round of 16 first-leg clash against Real Madrid on Wednesday at the Santiago Bernabeu Stadium.
Pep Guardiola has insisted mentality and personality would define Manchester City's season run-in after his side powered into the FA Cup quarter-finals with an impressive 3-1 win over Newcastle United.
LaLiga leaders Barcelona needed a moment of magic from Lamine Yamal to overcome to secure a 1-0 win over Athletic Club and protect their four-point lead over Real Madrid at the top of LaLiga standings.
Real Madrid coach Alvaro Arbeloa has insisted his side would continue fighting for the La Liga title despite recent setbacks, as pressure grows on the former defender following two damaging league defeats.
