The 13th-seeded American, who had been scheduled to team up with sister Serena, was close to tears as she talked about the injury that also saw her skip her only warm-up tournament in Auckland two weeks ago.
Her win over Swiss 26-year-old Voegele, ranked 112th in the world, was achieved in one hour and 23 minutes and set up a third round meeting with Duan Yingying.
"I'm managing it," Venus told reporters of the elbow problem. "I was really disappointed not to be able to play in New Zealand.
"I was hoping to have the capacity to play both events here, but at this point I just need to be careful and just try to maintain myself."
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Williams is making her 17th appearance at the Australia Open, where she reached the final in 2003. She was knocked out in the first round in 2016 by Britain's Johanna Konta.
The world number 17 will have disappointed fans who had hoped to see a repeat of the sisters' winning performance at Wimbledon last year.
The duo have taken home the Australian Open doubles title four times but the last one came in 2010.
The 36-year-old, the oldest player in the women's draw, was asked in a courtside interview about the changes she has seen in tennis over the last two decades.
"I have to talk about this every interview," she joked.
"I’ve played against some of the greats, Steffi Graf. I’ve even had a chance to play against Martina Navratilova so it’s been an honour and a privilege to I guess start that young and, play this old," she added laughing.
"This is the big leagues, we get out here and it’s all focus but at the end, you have a moment to enjoy the crowd. I love it, I got to be honest, it’s true love."
(Editing by Peter Rutherford)
