Therefore, he will face a runoff election in June against Republican Karen Handel, the no. 2 vote-getter.
Mr Ossoff and Ms Handel will now participate in a June 20 runoff.
Trump jumped into the race in its final days with attacks on Twitter poking fun at Ossoff for not living in his own district and for attracting support from Hollywood celebrities.
Ossoff has pitched himself as a check on Trump and an opponent to Washington gridlock.
When asked whether Spicer was concerned that "Ossoff came so close to 50 percent", Spicer said, "Well, again, I would - just looking at the facts, there was one candidate on the Democratic side".
Democrats heavily pushed Ossoff before Tuesday's vote, casting the race as a referendum against President Trump in hopes of turning the reliably red district blue. Nevertheless, Mr Ossoff's strong showing offered proof of a re-energised Democratic base that could fire up a a torrid fight for the Congress mid-term elections next year, said analysts.
Handel, Georgia's former secretary of state, won 19.8 percent of the vote.
The election has been closely watched throughout the country and is widely considered to be an early popularity test for President Donald Trump .
After the election, he tweeted: "Despite major outside money, FAKE media support and eleven Republican candidates, BIG "R" win with runoff in Georgia". If she can't, then Ossoff has a chance of winning in June. The Dunwoody retiree and Handel volunteer says Republicans were too cautious in the primary for fear they'd "step on toes".
"The reaction has been that they nearly won - no, they lost", Mr Spicer insisted.
DuBose Porter, chairman of the Georgia Democratic Party, said the district offers the flawless setting for Democrats to prove their momentum heading into 2018. "I think in large part, that's why we won", Spicer said. But he fell short of the majority needed to win outright. "We have defied the odds, we have shattered expectations", Mr Ossoff told a cheering crowd of supporters.
The Democrat led a crowded 18-candidate field to replace Health and Human Services Secretary Tom Price.