Ukraine were well beaten 3-0 by Romania in their tournament opener and when they fell a goal behind again in Duesseldorf, fans may have been fearing another defeat in a finals that it is hoped will raise spirits back home in the war-torn country.
"We made some mistakes when we conceded the goal and we had to react in the second half," Rebrov told reporters.
"I said to the players, ‘This is our chance. Don't worry about the score, you just have to give maximum effort on the pitch'. I'm pleased for them, they did that.
"I cannot say in the first game they didn't understand why they are here and who they are representing - the people who are fighting for the freedom, not only Ukraine but all of Europe.
"But they gave their all on the pitch today. They will celebrate a little bit tonight, but tomorrow is another training session."
Rebrov believes his side were not as poor as their opening loss suggested but, with their backs to the wall again, it became a mental battle as much as a physical one.
"It is great for the players that they showed the spirit of Ukraine," he said. "Of course, we have to concentrate on the next match (against Belgium on Wednesday) but today it was a very important win for our country, for Ukraine, for our fans, who I am sure today are very proud of the players."
Rebrov's side had 66% possession against Romania and lost but with 45% against Slovakia, they won and created more shots on target with five compared to two against the Romanians.
"The play at the Euros is very fast and you have to react quickly. Today we created more chances because we had a more wish to create chances. I think this is very important," he added.
"It doesn't matter that we controlled the ball against Romania, we lost 3-0. Football is about the score and about how many chances you create."