The 27-year-old, who signed from Liverpool in July for £75m, set up his new side's goal before being substituted on 69 minutes.
"It was a big relief. I'm very happy with the minutes I had," he said.
"I have a bittersweet sensation because of the result but this team has picked itself up before many times."
Barcelona won their opening eight league games while Suarez served his suspension for biting Italy defender Giorgio Chiellini at the World Cup, but their defeat in El Clasico leaves them just one point ahead of Real at the top of the table.
Some Madrid-based media had speculated that Suarez had put on weight during his enforced absence, but the Uruguayan insisted he was at full fitness.
"I felt really good. I have been training for a long time with this game in mind and so when the coach made the decision I was in the right shape to play," he added, in an interview with Canal Plus.
|European football writer Andy West|
|"The controversial Uruguayan made a superb start to his Barca career with an assist for Neymar's opener and also played a wonderful cross that Messi should have converted. Understandably faded and was replaced by Pedro."|
|Read the rest of Andy's player ratings|
"We went from having little control in the first few minutes to taking the lead but they are very good on the counter-attack and they killed off the game."
Suarez's cross-field ball picked out Neymar in space to score the game's first goal on four minutes, but a Cristiano Ronaldo penalty, Pepe's powerful header and Karim Benzema's cool finish at the end of a slick team move ensured Real saw out the match in comfort.
Barca coach Luis Enrique placed blame for the defeat on his defence, which had not conceded a single goal in their first eight league games but had been reorganised since their Champions League win over Ajax on Tuesday.
With Jordi Alba and Marc Bartra left out, Gerard Pique, Javier Mascherano and Jeremy Mathieu played together for the first time.
He also defended his decision to play Suarez from the start in the striker's first competitive match since June.
"I don't regret that he started, he did more than I expected him to do, and I substituted him for fitness reasons," Enrique said of Suarez.
"I was expecting a different type of game, not one in which we would be so unconvincing in defence.
"We have to correct that because if we don't we will really struggle - like we did today."