Venus, Lucie Safarova win in Dubai
The 48th-ranked Williams made her first appearance in Dubai since she won her second straight title here in 2010. She'd missed the last three years either because of injury or Sjogren's syndrome.
"I definitely don't take for granted any win anymore," Williams said. "Not that I used to because I earned it.
"But now it's even more special."
She fired 10 aces and broke the 33rd-ranked Vesnina four times. Vesnina had won their previous two matchups.
"My serve keeps improving pretty much every tournament," Williams said. "That always helps me."
In the first set, Williams broke Vesnina at love in the sixth game. Williams raced to 3-0 in the second set but surrendered her serve in the fourth game when she sailed a forehand long on the one break point she faced all night.
Williams' sister, Serena, the top seed and also a wild-card recipient, will play Ekaterina Makarova of Russia on Tuesday.
It will be the first match for Serena since losing to Ana Ivanovic in the Australian Open fourth round last month. Serena's been nursing a back injury until this week.
Safarova broke Stephens three times in the first set, but double-faulted twice from 30-30 in the ninth game of the second to give Stephens a 5-4 lead. Stephens, however, was unable to hold on to it.
"I didn't really get a great rhythm and she was playing good tennis in the end," Stephens said.
The main draw is so strong in Dubai that Australian Open semifinalist Eugenie Bouchard of Canada was relegated to the qualifying draw.
Bouchard lost her third qualifying match to Annika Beck of Germany 6-1, 6-4 on Monday.