Mina netted once in either half to secure all the points for Los Che, who have moved to within two points of third-placed Real Madrid in the La Liga table.
Valencia should have taken the lead in the 18th minute when Sociedad goalkeeper Antonio Ramirez parried a long-range strike from Francis Coquelin, but Rodrigo could only turn the rebound wide of the post.
Ramirez then made a smart save to keep out a Dani Parejo free kick in the 23rd minute, but the pressure was building from the home side, and the breakthrough arrived in the 34th minute.
Smart work from Martin Montoya set up the opener for Mina as the centre-forward had the simple task of heading home his teammate's cross from close range.
Sociedad had four attempts in the first period, but were not able to test the home side's goalkeeper Neto as Mina's goal separated the two teams entering the half-time interval.
The visitors were level in the 54th minute, however, when Mikel Oyarzabal took advantage of some slack Valencia defending to slot the ball into the back of the net with a calm finish.
Valencia regained the lead in the 68th minute though when Mina pounced on a loose piece of goalkeeping from Ramirez to register his second goal of the match.
Sociedad thought that they had levelled in the 79th minute when Sergio Canales's free kick headed for the bottom corner, but Neto made a fine stop before the Valencia goalkeeper also saved the rebound from Jon Bautista.
Former Manchester United attacker Adnan Januzaj then missed the Valencia crossbar with a curling effort in the 84th minute as the hosts were forced deeper and deeper in the latter stages of the clash at the Mestalla.
Four additional minutes were signalled at the end of the 90, but Sociedad could not find a leveller as they lost for the 12th time in La Liga this season, which has left them down in 14th position in the table.
VALENCIA (4-4-2): Neto; Montoya, Gabriel, Murillo, Gaya; Soler (Zaza 66'), Parejo, Coquelin, Guedes (Torres 73'); Mina, Rodrigo (Maksimovic 81')
REAL SOCIEDAD (4-3-3): Ramirez; Odriozola, Elustondo, Llorente, De La Bella; Zubeldia, Illarramendi, Zurutuza (Orgilles 79'); Canales, Juanmi (Januzaj 69'), Oyarzabal