With sister Serena watching on from the sidelines after being selected for the team, the American veteran proved too strong for Rus as she put the home side 1-0 ahead.
The Czech Republic, winners in five of the last seven years, ended day one of their tie against Switzerland 2-0 ahead after Petra Kvitova beat Viktorija Golubic in three sets before Barbara Strycova eased past Belinda Bencic.
Venus Williams, 37, was a big favourite against world number 138 Rus but was still relieved to claim her 776th career win.
"It's never easy and it's always a challenge," Venus said.
"On paper, today I looked like I should win but it was a battle and I'm glad to get through that."
Twenty three-times grand slam champion Serena has not played a competitive match since winning the 2017 Australian Open, since when she has become a mother for the first time.
She is pencilled in for doubles action.
Elsewhere, Belarus were locked at 1-1 with Germany while France and Belgium were also level going into Sunday's action.
Elise Mertens, an Australian Open semi-finalist last month, gave Belgium the lead with a 6-2 6-1 victory over Pauline Parmentier but Kristina Mladenovic beat Kirsten Flipkens 6-2 6-4.