Ms Solberg described her win as "a historic election victory for the right-wing parties".
Her Conservative Party is widely expected to form a government with the anti-immigration Progress Party.
It was the first election in Norway since Anders Breivik killed 77 people in a bombing and shooting in 2011.
With three quarters of the votes counted, the bloc of four centre-right parties had won 96 of 169 seats in parliament.
Mr Stoltenberg said he would step down after presenting the budget next month.
"In accordance with Norwegian parliamentary tradition, I will seek the resignation of my government after the presentation of the national budget on October 14, when it's clear that there is a parliamentary basis for a new government," he said.
The Labour leader, prime minister since 2005, congratulated Ms Solberg.
He said his party tried "to do what almost no-one has done, to win three elections in a row, but it turned out to be tough".
Ms Solberg, 52, will become Norway's second female prime minister and its first Conservative prime minister since 1990.
She told supporters in Oslo: "Today the voters have produced a historic election victory for the right-wing parties. We will give this country a new government."
Correspondents say at least the top two cabinet posts - and possibly the top three - are likely to be filled by women.
The vote was Norway's first general election since the attacks by Breivik - a far-right extremist - killed 77 people at a Labour Party youth camp in 2011.
Breivik had previously been a member of the Progress Party.
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