Madrid won 2-0 in Wednesday's second leg at the Santiago Bernabeu to secure a 5-1 aggregate success over their fierce rivals, Marco Asensio and Karim Benzema getting the goals.
Asensio's stunning fourth-minute strike set the tone and Carvajal felt Barca had no answer as Madrid added domestic silverware to the UEFA Super Cup crown they picked up last week.
"Our aim was to win both Supercopas and I think we were better than Manchester United and Barcelona," Carvajal told reporters.
"Our first 20 minutes were fantastic, we were all over them. We weren't affected by the result in the first leg. We were more clinical than them and that has been decisive in the tie.
"That gives us confidence going into the first LaLiga games. We have got a great squad and we are not surprised by what Asensio is doing."
Despite his confidence, Carvajal will not rush before declaring Madrid can win a sextuple this season.
"There is still a long way to go this season, we have barely begun. We will go week by week and, come March, we will see whether we can win it all," he said.
“But I don't see Barca as far away, they had chances to win too."
Lucas Vazquez, meanwhile, wants Madrid to keep up the momentum for their defence of LaLiga, which begins on Sunday away to Deportivo La Coruna.
"We are all thrilled to have won this trophy, which gives us confidence heading into the LaLiga campaign," he said.
"I think we were the better side in both games. We have got a fantastic group and we are in a great dynamic.
"The secret to this team is that whoever is out on the pitch, they do a great job and we all give it everything we have got to defend the badge."