The Eibar talisman struck twice in the victory - only their second away from home this season - as they showed fight to come back and rescue all three points at Estadio del Rayo Vallecano.
Arruabarrena gave the visitors the lead from the penalty spot on 37 minutes, and when Federico Piovaccari doubled their lead four minutes after the break victory looked assured.
However, two goals in a minute from Leo Baptistao brought Rayo right back into the game.
The striker's first was a neatly guided header from Lica's left-wing cross, and straight from the kick-off he doubled his tally, converting a low cross from Gael Kakuta, who is on loan from Chelsea.
Arruabarrena was not to be denied, though, the 31-year-old meeting Abraham Minero's whipped cross with a sumptuous volley into the top corner, giving David Cobena no chance.
The win lifts Eibar back into the top 10, while Rayo slip down to 12th after a second defeat in a row.
