The Australian, playing her first match of the US hardcourt season after having received a first-round bye, dropped to a shock 6-2 6-4 defeat to her 83rd ranked opponent from Belarus.
Stosur's outing at the tournament in California was serving as a warm-up for her tilt at the upcoming US Open, which she won in 2011.
But Stosur's current form is a far cry from that which landed her a first and only singles Grand Slam title two years ago, and Govortsova was able to run away with victory.
The Belarusian won the first four games of each set while Stosur double-faulted 10 times.
The Australian's attempted fightback late in the second was too little too late and Govortsova sealed victory to ease into the third round, where she will now meet either Sorana Cirstea or Coco Vandeweghe.
Top seed Agnieszka Radwanska has a tough opening match against Francesca Schiavone in the second round.
Stanford Classic results
Olga Govortsova (Belarus) beat 2-Samantha Stosur (Australia) 6-2 6-4
7-Urszula Radwanska (Poland) beat Christina Mchale (U.S.) 6-1 6-3
Daniela Hantuchova (Slovakia) beat Yanina Wickmayer (Belgium) 6-2 4-6 6-0
Vera Dushevina (Russia) beat Marina Erakovic (New Zealand) 6-2 6-1
Coco Vandeweghe (U.S.) beat Monica Niculescu (Romania) 6-0 6-3
Tamira Paszek (Austria) beat Alla Kudryavtseva (Russia) 6-1 6-7(4) 3-0 (Kudryavtseva retired)
6-Varvara Lepchenko (U.S.) beat Michelle Larcher de Brito (Portugal) 6-2 6-4
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