Pellegrini's Real Betis had won their opening two matches without conceding, but were picked apart defensively at the Ciutat de Valencia.
"The defeat is easily explained because defensively we were a disaster," the Chilean said. "It was a fairly even first half - they came at us, we came at them, we finished it 2-2, also with errors from our defence on their goals. We started the second half with the ball at 2-2 in Troy Parrott's six-yard box, and they scored three against us.
"They could have scored two or three more, and we barely reached their goal, so unfortunately what we've been doing - including the defensive work - today I don't think we had the concentration or the intensity to win this match."
Pellegrini rejected suggestions that fatigue from early-season rotation was to blame.
"I don't think so, because we've made a fair number of rotations given it's the start of the season and we came off a pre-season. We rotated quite a bit so the team would respond physically. I think it was more an individual performance issue - mistakes were made, especially on set pieces, where you can't afford to give that advantage."
He did find one positive: "We managed to come back twice in the first half from behind on the scoreboard. We had the attacking capacity to pull it back to 2-2. I think that was positive, but honestly if the first half was poor defensively, the second was much worse."
For home boss Luís Castro, the win over a side already qualified for Europe underlined the progress made since his side's opening-day defeat to Espanyol.
"We knew we would be stronger, because here we have enough quality. In the last match, the team grew a lot. It's only three points, though we have to enjoy it because we played against a Champions League team," the Portuguese said. "We controlled the ball, we were intelligent and we knew how to suffer, but it's only three points. We have to keep working."
Castro also praised his side's collective spirit over individual quality, and credited a settled home crowd for six straight wins at the Ciutat: "Before we didn't get points here, but in this stadium there's a very good atmosphere. They're always with us. If we're together, it's easier."
On the market uncertainty facing a selling club like Levante, he added: "I don't look at problems, I look at solutions. But here, as I always say, whoever isn't at 100 per cent doesn't play."