Sevastova is no grasscourt rookie, having recently won the Majorca Open but she had no answer to Watson's all-round game in a first set which lasted just 19 minutes.
Two-time Grand Slam victor Azarenka beat 15th seed Elena Vesnina to join Watson in round three, in her first return to Grand Slam action since giving birth to her son Leo in December a year ago.
While Watson stopped short of empathising too much with Bernard Tomic after his claim that he was bored during his first-round match, she has experienced her own struggles with motivation. "More relaxed than in my first match. I was pleased with how I remained calm and just kept doing what I was doing". "I made those big goals years ago".
Azarenka said: "I think she's a great player". "And although I hit a few double faults, I feel like I'm serving very well".More news: GST: The Goods and Services tax enrolled from today
Five-times Wimbledon champion Venus Williams came back from a set down to beat China's Wang Qiang 4-6 6-4 6-1 on Court One.
There was a win for eighth seed Dominika Cibulkova on Court 18 who came through 6-4 6-4 against Jennifer Brady of the US.
There was an upset for Barbora Strycova, seeded 22nd, who was defeated by Japan's Naomi Osaka who is ranked at 59 in the world.
She then rose to a career-high mark of 38 in January 2015 before slipping down again.
Heather Watson is within touching distance of achieving her long-term goal of reaching the second week of a grand slam - which could be bad news for coach Morgan Phillips.