Raphinha scored twice either side of a first-half Fermín López double, with Karim Adeyemi and a second Fermín strike completing the rout at the Martínez Valero, though the afternoon was overshadowed by Gavi limping off before the break after a knock to his right knee.
Flick was pleased with the control his side showed in a lopsided contest.
"First, we needed to control the ball a bit more, and that's what I want from the team. We managed to keep a clean sheet and score five goals. I love the way Elche play football."
On Gavi, he was cautiously optimistic despite ordering scans.
"It's the knee and we have to wait. He's better than I expected after the moment of the injury. We'll do an MRI."
Asked about Barcelona's search for a new number nine, Flick kept his answer brief.
"We'll talk after the match. I speak with Deco and we'll see what we can do. We'll see next week. We're happy to have scored five goals. It gives us confidence for the games ahead."
On Alejandro Balde's continued absence from the squad, he added: "I haven't told him to leave. Something is going on with him, I can see it. I spoke with him this morning and I have that feeling. It's best that he clears his mind and we'll talk tomorrow."
Flick praised the quality shown as the game opened up in the second half.
"I've seen a lot of quality in training. The most important thing is to improve in training, and if you have that quality, you can show it in matches. We played as a team, we attacked and defended together."
He singled out academy product Xavi Espart, deployed at left-back, "He played with the under-19s, he's in form, his level is very good. I love seeing a La Masia player at a great level again. It doesn't matter if he plays right, left, central... he's fantastic."
Raphinha also drew praise for his attitude in training: "He has a great mentality. When you see players give one hundred per cent in training, it's fantastic. He's a good example."
Elche coach Martín Anselmi, making his home debut in charge, said the scale of the defeat made words feel almost beside the point.
"After this kind of defeat, the weight of the result means words are unnecessary. What we can take from it I'll have once I review the match. When we competed together, we were able to cause damage. And they knew how to strike. The boxes are decisive. You can't give them any opening. With it at 0-2 and 0-3, we had some situations to get back into the game."
Asked whether the second goal before half-time had proven decisive, Anselmi agreed it changed the complexion of the match.
"It comes at a moment when I felt our goal was coming. We had situations, like a set piece we couldn't convert, Ali's chance, organised attacks... we felt we could score at any moment. But we paid for our mistakes. With it at 0-2, the match becomes a different one."
He also reflected on the emotion of his first home game in charge.
"Arriving at the stadium and seeing how many people were there, when we stepped onto the pitch... it was hard to contain myself, it stirred a lot of emotion inside me. In the final result, we didn't represent the supporters. The result damages the image we wanted to give. We're grateful to the fans for being there. We want them to keep coming."