The 51-year-old described the match, which finished 2-2 after extra time, as "a battle" that could have gone either way.
"Congratulations to my lads but also to the Russia team, who fought well. This was not a beautiful game, it was a fight, a battle for the semi-final. We were lucky," he told reporters during his post-match press conference.
"I was concentrated throughout the match but after Rakitic's penalty, it came bursting out of me. I felt really relieved. We made ourselves happy but we also made everybody back home in Croatia happy. Those emotions just burst out of me. I don't cry often but now I have a good cause: Croatia are in the semi-finals. This is a great success and a great result.
"Of course there is some power left for the English. We do not want to stop, we will try to play our best game. We have two matches to play and we are very motivated. We're giving our all. It will be a battle again but I trust us, I have belief in us."
England will meet Croatia in the last four after seeing off Sweden 2-0 in the quarters.