He will face Republican challenger Karen Handel, a former Georgia secretary of state who attracted 19.8 percent of the vote.
"National Right to Life is pleased to endorse Karen Handel, whose position on life reflects the true values of Georgia's voters", said Karen Cross, National Right to Life political director.
As they see it, a win in Georgia could jumpstart efforts to retake the House in 2018.
Despite what seemed like at least a moral victory for Mr Ossoff, the White House claimed that it was in truth a defeat.
Mr Price had won the seat in the 2016 election by a margin of 23 points - something Mr Ossoff, a political neophyte hopes he can turn around in the run-off against Ms Handel. "It is now Hollywood vs. Georgia on June 20th", Trump wrote, alluding to celebrity donors to Ossoff. Handel outperformed some of her polling numbers and finished a strong second, leaving 11 other Republicans, some of whom spent more ad money, behind.
That attention has unnerved some Republicans, including President Donald Trump, who attacked Ossoff via his Twitter account.
"Only the Republicans could try to define this as a defeat for the Democrats", said Porter. "If we can get over 50 percent in this district, we know we can do that statewide and around the country", Porter said, mocking Republican assertions that Ossoff failed Tuesday by falling short of an outright primary victory. "They came up short". Should they keep plowing millions into this race?
"Obviously, we're very excited, and what happened last night is very encouraging", said Handel.
"I think it's overstated the effect that what is going on in Washington has on a local race like this, where local issues are the most important thing to voters", Ossoff said.
"We defied the odds and shattered expectations - and we are ready to fight on and win in June".
While Democrats can savor the moral victory of almost snatching a seat outright in a Republican stronghold, they will need to translate anti-Trump fury and substantial fundraising into concrete ballot-box results if they are to mount a viable comeback.
Associated Press reporter Kathleen Foody contributed.