Madrid looked to be heading for three points in their first La Liga match of 2019, with Raphael Varane and Karim Benzema putting them ahead at Estadio de la Ceramica after Santi Cazorla's fourth-minute opener.
However, Cazorla struck again with eight minutes of normal time remaining to ensure the game finished 2-2, meaning Santiago Solari's side could only close to within seven points of league leaders Barcelona.
Vazquez admits they are unhappy with the result but remains confident in their trophy chances, especially since they won the Club World Cup under two weeks ago.
"We head home angry because it was a game we wanted to win and we let it slip with a lapse in concentration," he said, as quoted by Madrid's official website.
"The game was going how we wanted it to at 2-1, and they levelled with 82 minutes gone. We didn't have time to react and we go away with a bitter taste in our mouths, having dropped two points.
"We struggled to play through their press in the second half and so we were sat deeper, and they got a chance and scored their goal.
"Given this team's recent history, we deserve to retain people's faith in us. We're going to give it our all like we do every year and come the end, I'm sure we'll achieve great things. We've got another game on Sunday [against Real Sociedad] to do the best we can."