The hosts bossed the first half and deservedly took the lead six minutes before half-time through former Espanyol player Coro, but Espanyol were rejuvenated after the break, leading to Thievy Bifouma pulling them level on 65 minutes.
The visitors looked as if they would go on and win the game as they continued to press Manu Herrera's goal, but Coro got his second goal with little more than a quarter of an hour remaining, which was enough to deliver a 2-1 win and all three points.
Espanyol were reduced to 10 men in the closing stages when Victor Sanchez picked up his second yellow card of the game.
Elche went into the match after finally getting their first win since returning to the Primera Division, beating Celta Vigo 1-0 last weekend at Balaidos.
They looked intent on building in that victory from the start, closing down Espanyol in midfield and putting pressure on the visiting goal.
Kiko Casilla had to make a smart save to prevent Coro giving his side the lead before the half-hour mark, and moments earlier blocked a powerful albeit central effort from Manu del Moral.
The goal eventually arrived when Coro received the ball in the box following a throw in, turned a defender and then shot past Casilla.
The Catalans showed renewed effort after the break. Sergio Garcia tried in vain to equalise early in the second half with a back-heeled effort that missed the target, but the forward played a part in the eventual equaliser, combining with
Simao Sabrosa and Thievy on the edge of the area, allowing the Frenchman to shoot past Herrera.
Herrera then denied Garcia a certain goal at the near post, pushing the ball away for a corner.
Barely two minutes later, Elche went up the other end to score.
Coro received the ball just outside the six-yard box from the left-hand side and controlled it then poked it into the bottom corner.
Espanyol's efforts to get an equaliser took a blow when Sanchez received his second booking in 15 minutes, the latter after bringing Ruben Perez down from behind.