La Liga wrap: Olinga, 16, creates history in Malaga win
With Santi Cazorla and Salomon Rondon already departed for Arsenal and Rubin Kazan respectively, the Cameroon-born striker was thrust into the action by Manuel Pellegrini on the hour mark and he netted the decisive goal in the 84th minute.
Celta had a chance to snatch a draw in the dying moments but Gustavo Cabral's strike bounced off the bar.
Olinga's goal saw him become the league's youngest goal scorer at just 16 years and 98 days, eclipsing the previous record set in 2009 by Athletic Bilbao's Ilker Muniain, who was 16 years and 289 days old.
Meanwhile, a brace from Israeli striker Tomer Hemed inspired Mallorca to a 2-1 victory against Espanyol.
Mallorca, who had lost the pair's most recent encounter last season, hit the front inside the opening two minutes of play through Hemed.
However, the visitors were on level terms four minutes later after Wakaso Mubarak scored his first goal for the club.
The match looked on course for a draw but Hemed broke free inside the penalty area and smashed home the winner on 85 minutes.
In Saturday's other result, Sevilla recorded a 2-1 win over Getafe on the fifth anniversary of Antonio Puerta's death.
The former Sevilla and Spanish international collapsed on the field in this same fixture five years ago and later died of a heart attack, aged 22.
Federico Fazio and Alvaro Negredo both scored in the space of four minutes to give Sevilla a two-goal lead at the Estadio Ramon Sanchez Pijuan.
Alexis headed home a 50th-minute goal for the visitors but it proved nothing more than a consolation.