Los Che were left ruing a number of chances in the first-half as they succumbed to their second home defeat while the resolute Basque side move further clear of the drop zone
In an otherwise lacklustre match, the Frenchman put Sociedad ahead when he reacted brilliantly to his own rebound lifting the ball over a stranded Diego Alves.
Los Che, determined to find an equaliser, attempted to break through the dogged visiting defence but Claudio Bravo's goal remained impenetrable as the league strugglers hung on to a surprise victory.
In a frantic start, Valencia wasted numerous opportunities to take the lead, with Roberto Soldado's replacement Artiz Aduriz unable to convert his chances.
Valencia's top scorer was suspended for the visit of Sociedad, and Los Che missed his impressive strike rate as his deputy was denied twice by Bravo in the opening moments.
Aduriz did manage to beat the goalkeeper when he directed Dani Parejo's corner toward goal but it clipped the top of the crossbar on its way behind.
The Basque side succeeded in escaping the first half hour unscathed and grew in confidence as the half progressed. However, Philippe Montanier's side were unable to create much in the way of chances despite pressurising Valencia's goal.
Unai Emery's side returned from the break with a fresh impetus as both Pablo Hernandez and Sofiane Feghouli had early efforts on goal.
With neither side able to find the piece of creativity that they needed to force an opening, Real Sociedad launched an emphatic counterattack with Griezmann smartly lifting the ball over Alves into the back of the net.
Valencia attempted to find an immediate response but Sociedad were defending resolutely as they continued to threaten on the counter to prevent an attacking onslaught from the home side.
Griezmann came close to doubling the visitors' advantage as he took aim from distance but his swirling effort, which had Diego Alves scrambling, went wide of the target.
Los Che strained to find a late equaliser and the home fans were left frustrated as a seemingly below-par performance ended in defeat. Valencia face Levante next in their Copa del Rey quarter-final tie in midweek and will be hoping to rediscover their scoring form, which had seen them net in all but two marches before tonight.
Montanier's side meanwhile host Atletico Madrid next weekend as they continue to escape the clutches of relegation.