Marc Bartra's header cancelled out Angel di Maria's opener at Mestalla on Wednesday, but a fine individual goal from Gareth Bale saw Real secure a 2-1 success.
Barca exited the UEFA Champions League courtesy of a 2-1 aggregate defeat at Liga leaders Atletico Madrid last week, while a 1-0 setback at Granada on Saturday has severely dented their hopes of defending the Spanish title.
Defeat to Real therefore marked a third consecutive disappointment and added to the pressure mounting on Barca head coach Gerardo Martino.
However, Iniesta has urged the team to fight on in a bid to overhaul Atletico and Real in the league.
"It's a shame. We have to be thankful to the fans, they were with us all the way through to the end," he told the club's official website.
"We're coming off of a week in which we've been eliminated from two competitions. We tried until the end and details make a difference.
"We're leaving here disappointed that we didn't win the (Copa) title. You try to have the initiative in the game but Real Madrid are very decisive when they counter-attack. We have to find strength to fight for La Liga."
Fellow midfielder Xavi agreed that Real's speed on the break was decisive.
He added: "They hurt us with their counter-attacks. If anyone is at fault, it's the team. From the manager to the players. These are our errors, and our rival took advantage of them.
"We still have La Liga, and it will be difficult to win it, but we will continue fighting for the title."
