Real's dramatic 1-0 victory in Valencia, courtesy of Cristiano Ronaldo's 103rd-minute header, ended Barca's hopes of a second treble in three seasons but they are in a strong position to claim a third straight Spanish league title with six games left.
They host struggling Osasuna on Saturday before their clash with Real in the two-legged Champions League semi-final on April 27 and May 3.
"We had hoped for victory (in the Cup final) but we have to pick ourselves up and digest this because we have a month left that could be very nice," Villa told reporters after the match at the Mestalla stadium.
"We have to clinch the league and think about the Champions League where we have the chance to reach the final," he added. "There are some very nice experiences ahead."
Barca coach Pep Guardiola, who tasted his first defeat in a final since taking charge at the end of the 2007/08 season, said he was confident his players would demonstrate the same resilience they had shown in the past.
"It is always better to win than lose that's clear," he said at a news conference.
"We will come back from this, of course. These players have showed they can do that many times, with me and without me."
Jose Mourinho's Real are on a high ahead of Saturday's game at third-placed Valencia after claiming their first silverware in three seasons and their first King's Cup since 1993.
It was also their first trophy under Mourinho, who has now won domestic cups in four countries after previous triumphs in his native Portugal, England and Italy.
PSYCHOLOGICAL BOOST
Former Real player Zinedine Zidane, an adviser to club president Florentino Perez, said defeating Barca would give the players a confidence boost but appeared to write off the club's chances of catching Barca in the Spanish league.
"This victory is very important on a psychological level," the former France international said on Real's website.
"This triumph is important, but we must now focus on the Champions League," he added.
The fight for a place in next season's Europa League has been hotting up in recent weeks and only six points separate Athletic Bilbao in fifth and Levante in ninth, with Sevilla, Atletico Madrid and Espanyol in between.
Fourth-placed Villarreal, nine points clear of Bilbao, play at Sevilla on Sunday and Bilbao host Real Sociedad on Saturday.
Atletico are at home to Levante on Sunday, when Espanyol travel to Gijon to play Sporting.
At the other end of the table, bottom club Almeria are at fellow strugglers Real Zaragoza on Monday and Hercules, in 19th place, host Deportivo La Coruna on Sunday.
