The British number one, seeded fourth in Memphis, won an error-strewn clash 7-5 5-7 7-6(4) in a whopping three hours and 24 minutes to move into the second round.
Watson, the world number 41, faced an incredible 18 break points but saved 11 of them; Kazakhstan’s world number 109 Voskoboeva was also broken seven times, having faced 11 break points.
Watson’s risky service game – she managed eight aces and six double faults – was countered by some better returns, but she was far from her best form.
The Channel Islander was 5-2 up in the third set before allowing herself to be pegged back to a tie-break, but she managed to grind out the win.
Watson next faces the winner of Melanie Oudin’s match with Andrea Hlavackova.
The top seed is world number 34 Kirsten Flipkens, with Sofie Arvidsson (38) and Sabine Lisicki (40) the others ranked above Watson in what is a winnable tournament for the Briton. They all play on Monday.
Another Briton was in action on Sunday, as Watson's Fed Cup team-mate Johanna Konta lost 6-3 6-7 (6-8) 6-3 to Croatia's Tereza Mrdeza in Bogota, Colombia.
WTA Memphis results (round one):
4-Heather Watson (Britain) beat Galina Voskoboeva (Kazakhstan) 7-5 5-7 7-6(4)
Coco Vandeweghe (U.S.) beat 6-Chanelle Scheepers (South Africa) 6-2 6-1
Lauren Davis (U.S.) beat Ksenia Pervak (Kazakhstan) 5-7 6-1 6-3
7-Magdalena Rybarikova (Slovakia) beat Stephanie Foretz Gacon (France) 6-3 6-1
Melinda Czink (Hungary) beat Olga Puchkova (Russia) 4-6 6-3 6-3