Fer Niño scored twice in the space of three minutes, in the 26th and 29th, with Roy Revivo supplying the final pass for the second.
Elche arrived at the RCDE Stadium bottom of the table and still without a win, and the sending-off of Federico Redondo, after a VAR review for hauling down an Espanyol winger who was about to go clear, had left them short-handed inside the opening minutes.
Víctor Chust put the ball into his own net on 72 minutes to give the hosts hope, but Germán Valera restored the two-goal cushion six minutes later with a superb individual finish.
Elche coach Martín Anselmi, quoted by Marca, was quick to spread the credit.
"I'm proud of everyone. If Roy doesn't cross it, Fer doesn't score. Football is a collective sport and it depends on everyone. There's always some individual who stands out, or not, but the star is the team."
He felt his side coped intelligently with the disadvantage.
"During the week, I had said I wanted Elche to be intelligent and competitive. That is adapting to the circumstances of the game, and as we were intelligent and we competed, we developed our game. It gives me satisfaction. I am extremely happy with how we played today."
Anselmi also dedicated the win to his family: "It took a while to arrive, but more than a personal relief, it is also one within the club. It's the effort of my wife and my children. This triumph is for them."
The mood in the Espanyol camp was very different. Manolo González, according to Marca, called it the worst performance of his time at the club.
"It was an awful game. They put us in a very good situation to win and, from that moment, everything we could do badly, we did badly. I think it's the worst game since I've been at Espanyol and there's nothing more to say."
The coach put the blame on himself first.
"I'm quite clear about where I'm going wrong, I'm quite clear about it."
Beaten for a second consecutive match after losing to Rayo Vallecano, Espanyol now have almost three weeks to respond. González said the supporters' whistles were only natural.
"Of course the atmosphere isn't the best, but we've caused that ourselves. So when you cause the situation, what you have to do is reverse it. That's all there is. And we have to work to turn it around, there's no other option."