Aspas scored from the penalty spot on the hour to put Celta 2-1 up after Raul Albentosa had cancelled out Hugo Mallo's opener.
Fabian Orellana added a third for Celta in the 78th minute before Aspas, 29, sealed the win seven minutes from time.
Aspas scored once in 15 appearances for Liverpool in the 2013-14 season.
Liverpool signed Aspas from Celta for £7m in June 2013 but he struggled to make an impact and was sold to Sevilla.
He rejoined Celta for a second spell in June 2015 and scored 14 goals in 35 appearances in La Liga last season.
Match ends, Celta de Vigo 4, Deportivo de La Coruña 1.
Second Half ends, Celta de Vigo 4, Deportivo de La Coruña 1.
Attempt missed. Pedro Mosquera (Deportivo de La Coruña) header from very close range is too high. Assisted by Celso Borges following a corner.
Attempt missed. Celso Borges (Deportivo de La Coruña) header from the centre of the box misses to the left. Assisted by Faycal Fajr with a cross following a corner.
Corner,  Deportivo de La Coruña. Conceded by Pablo Hernández.
Attempt blocked. Florin Andone (Deportivo de La Coruña) right footed shot from a difficult angle on the right is blocked. Assisted by Carles Gil.
Nemanja Radoja (Celta de Vigo) wins a free kick in the defensive half.
Foul by Marlos Moreno (Deportivo de La Coruña).
Substitution, Celta de Vigo. Pione Sisto replaces Orellana.
Cabral (Celta de Vigo) wins a free kick in the defensive half.
Foul by Florin Andone (Deportivo de La Coruña).
Substitution, Celta de Vigo. Giuseppe Rossi replaces Iago Aspas.
Foul by Cabral (Celta de Vigo).
Florin Andone (Deportivo de La Coruña) wins a free kick in the attacking half.
Corner,  Deportivo de La Coruña. Conceded by Facundo Roncaglia.
Goal!  Celta de Vigo 4, Deportivo de La Coruña 1. Iago Aspas (Celta de Vigo) left footed shot from the left side of the box to the bottom left corner. Assisted by Sergio Álvarez.
Offside, Celta de Vigo. Pablo Hernández tries a through ball, but Iago Aspas is caught offside.
Nemanja Radoja (Celta de Vigo) wins a free kick on the left wing.
Foul by Carles Gil (Deportivo de La Coruña).
Substitution, Deportivo de La Coruña. Celso Borges replaces Guilherme.
Goal!  Celta de Vigo 3, Deportivo de La Coruña 1. Orellana (Celta de Vigo) header from the centre of the box to the bottom left corner. Assisted by Jonny with a cross.
Attempt saved. Orellana (Celta de Vigo) right footed shot from outside the box is saved in the centre of the goal.
Pedro Mosquera (Deportivo de La Coruña) is shown the yellow card for a bad foul.
Orellana (Celta de Vigo) wins a free kick in the attacking half.
Foul by Pedro Mosquera (Deportivo de La Coruña).
Substitution, Celta de Vigo. Marcelo Díaz replaces Theo Bongonda.
Substitution, Deportivo de La Coruña. Carles Gil replaces Ryan Babel.
Delay over. They are ready to continue.
Delay in match Hugo Mallo (Celta de Vigo) because of an injury.
Offside, Deportivo de La Coruña. Faycal Fajr tries a through ball, but Guilherme is caught offside.
Orellana (Celta de Vigo) is shown the yellow card.
Foul by Orellana (Celta de Vigo).
Fernando Navarro (Deportivo de La Coruña) wins a free kick on the left wing.
Attempt blocked. Pedro Mosquera (Deportivo de La Coruña) left footed shot from outside the box is blocked.
Corner,  Deportivo de La Coruña. Conceded by Sergio Álvarez.
Pablo Hernández (Celta de Vigo) is shown the yellow card for a bad foul.
Foul by Pablo Hernández (Celta de Vigo).
Marlos Moreno (Deportivo de La Coruña) wins a free kick in the attacking half.
Attempt missed. Faycal Fajr (Deportivo de La Coruña) right footed shot from outside the box is close, but misses to the left following a set piece situation.
Attempt blocked. Guilherme (Deportivo de La Coruña) right footed shot from outside the box is blocked.
Former Liverpool striker Iago Aspas scored twice as Celta Vigo beat La Liga rivals Deportivo La Coruna 4-1 in the Galician derby.