Real beat 10-man Atalanta 1-0 via Ferland Mendy's late winner.
Vazquez later said: "We knew it would be a tough game but I think it's a fair result and we deserved the win. They sat back after the red and they defended really well.
"They did what the had to and it was tough to find space but the goal we scored makes us happy. We're not taking anything for granted and know there's a second leg to come."
Real goalkeeper Thibaut Courtois added: "We were really well focused, the away goal means a lot but the return leg will be a different game. The Champions League is always tough."