Miroslav Djukic’s men had put together back-to-back wins to leave the relegation zone and Andone’s goal inside 10 seconds looked like boosting the promoted club towards the top half of the table.
Both sides lost a man to red cards in the second half, though, Eduard Campabadal and Federico Piovaccari’s dismissals destabilising Cordoba’s game plan and allowing Mikel Arruabarrena to notch a headed equaliser.
The point helps the Andalusians climb one place to 14th while Eibar remain in eighth position.
Cordoba may still have been riding the wave of enthusiasm generated by their Monday night victory at Rayo Vallecano but they will not have expected to hit Eibar so early in their next appointment.
There were only 9.8 seconds on the clock when Federico Cartabia’s ball forward was chested down by the bustling Andone and surprisingly flicked into the visitors’ net.
It was the fourth fastest goal in La Liga history – behind those of Joseba Llorente, Seydou Keita and Dario Silva – but Cordoba were not prepared to get caught cold celebrating with at least 89 minutes still to play.
Fausto Rossi soon shot straight down Xabier Irureta’s throat and goal provider Cartabia fizzed a speculative shot just wide of the mark on 15 minutes.
Piovaccari produced Eibar’s best half-chance of the half but nothing would touch Andone’s shock opener.
The odds were evened up, briefly, 10 minutes after the break as Campabadal, booked in the first half for fouling Eibar’s Manuel Lillo, impeded Eneko Boveda to earn his second yellow card and his marching orders.
However, Piovaccari soon followed him down the tunnel as flaring tempers saw the Italian go in hard on Andone.
Andone made good use of the extra space to cut into the box in the 71st minute, but his shot from close range was parried by Irureta.
Eibar capitalised on the wasted opportunity immediately, racing up the other end to score.
Saul swung in the ball from the right and Cordoba goalkeeper Juan Carlos was left flapping as Arruabarrena headed home from six yards out.
The hosts were visibly incensed having let their lead slip away and twice went close to restoring it before full-time.
First, Cartabia headed Abel’s free-kick over the bar and, in stoppage time, Jose Angel Crespo nodded the ball into Irureta’s gloves as Cordoba piled forward en masse.