When Dr Phelps takes up her seat, there will be a crossbench of six members in the House of Representatives with Labor holding 69 seats and the coalition one short of a majority with 75.
Prime Minister Scott Morrison described it as a "tough day" and pledged to "listen, learn and accept the blows".
However he appeared unfazed about having to negotiate a minority parliament, as Julia Gillard was forced to after the 2010 election.
"We will work constructively with all of those who sit on the cross bench as we always have," he said.
Dr Phelps said she would be setting two priorities - action on climate change and getting asylum seeker children off Nauru.
She may also lend support for a national integrity commission which has been promoted by Labor, the Greens and crossbenchers for some time.
Independent Bob Katter said he did not want to see the government destabilised or an early election called.
But he said the coalition needed to address the drought, regional development and indigenous health.
Fellow independent Andrew Wilkie said he would not guarantee confidence or supply.
"That will continue to be the case," Mr Wilkie said.
Another crossbencher Cathy McGowan declined to comment.
Disgruntled Liberal MPs pointed the finger at failed leadership contender Peter Dutton's boosters for the by-election loss.
"The blame for this lies squarely at the feet of anybody who felt it was a good idea to get rid of Malcolm Turnbull," an angry Liberal backbencher Craig Laundy said.