Carlo Ancelotti's side won the UEFA Super Cup on Tuesday and the experienced defender believes the ingredients are there for greater success this season.
Carlo Ancelotti's side has recruited three of the stars from the 2014 World Cup in Brazil ahead of the new campaign as Toni Kroos, James Rodriguez and Keylor Navas arrived at the club.
Real picked up its first piece of silverware on Tuesday as a Cristiano Ronaldo double gave Ancelotti's men a 2-0 win over Sevilla in the UEFA Super Cup in Cardiff.
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Both James and Kroos made their competitive debuts in the victory and wasted little time making an impact, with the German controlling midfield and the Colombian showing flashes of fine link-up play with Ronaldo and Gareth Bale in attack.
Pepe believes the ever-evolving Real squad is well balanced for the demands of challenging for silverware at home and in Europe.
"I think we are fine," said the Portugal international. "We have a team with plenty of quality and also with a lot of competence inside the team that it is good for Real Madrid. We have started very good this season and the team looks good.
"It could be a new era for Real Madrid but we have to keep working and respecting our rivals because they have plenty of quality and I hope the year will be fine for Real Madrid."
Real erased the disappointment of seeing Atletico Madrid win La Liga last season by beating its city rival in the Champions League final.