The win restored Real's four-point lead over Barcelona in the table with just two games left to play, meaning a home win on Thursday can put the title out of reach for the two-time defending champions.
"It's true that we're one step closer but we have to keep going, there are still more points to play for and we're going to try and win every game left as that was always our objective," said captain Sergio Ramos.
"I really hope we can celebrate winning this league title, our results speak for themselves."
The visitors wasted no time pulling ahead, with Ferland Mendy finishing off a nice individual move in the 10th minute by smashing his shot from a tight angle into the roof of the Granada net for his first Blancos goal.
"Without realising it, we made the mistake of letting our guard down after getting a scoreline which we would have been very happy with," Ramos said.
"But it wasn't good enough and today you can't beat anyone unless you are fully focused for 90 minutes. We won the first half and lost the second and although we got the victory we did things we should never do, like conceding a cheap goal."
It was the first goal Real had conceded in six matches but Granada threatened to equalise as Machis missed the target on the volley while Courtois showed superb reflexes to deny forward Antonin late on.
Real Madrid were made to work hard to see out the result and had Ramos to thank for sweeping a Ramon Azeez shot off the line in the 84th minute as Zinedine Zidane's side picked up a nervy three points.
Zidane celebrated wildly when the full-time whistle came and praised his side for coping with Granada's late onslaught.
"I let out a shout because I was happy and because in football you have to suffer," said the Frenchman, who is closing in on his second La Liga title in under four years as Real coach along with three Champions League crowns.
"I shouted out loud because the three points were very important and I wanted to tell my players that they had played a superb game. To win this league you have to suffer a little bit and we dug in as a team. You have to be proud of this team."
The result puts Madrid on 83 points after 36 games and gives them a chance to seal a first La Liga title since 2017 on Thursday against Villarreal.
"Winning the league would mean a lot to us, it would be the reward for the consistency we have shown and for how hard we have worked throughout the year," added Ramos, whose side were two points behind Barcelona when the season was halted in March due to the COVID-19 pandemic.
"The lockdown served us well and proved to be a turning point as we worked on preparing to return to action and set ourselves the objective of winning the title.
"Now we are in the title fight and I hope that on Thursday we can celebrate it properly."
Two draws in the last two games can also secure the title for Madrid due to their better head-to-head record with Barcelona.