Ms Davidson’s party won 31 seats, more than doubling its representation, marking the success of her keynote message to oppose a second referendum in Scotland during the next five years and provide a strong opposition to the SNP.
And with 24 of the party’s MSPs being newly elected, Ms Davidson says they will bring a fresh approach the Holyrood chamber.
Prime Minister David Cameron has hailed a “historic result” for the Conservatives after their success.
Ms Davidson pledged to “give a voice to all those who feel the government is not fully addressing their concerns”.
She added: “The SNP now needs to listen to the message delivered to them on Friday night.
“Yes, people in Scotland have asked them to continue in power. But people are demanding that the SNP use that power with greater care.
“They are demanding a government that seeks to listen, not to hector.”
The Tories will now focus on manifesto pledges including increasing college funding, supporting people suffering mental ill-health and “forcing change” to the controversial named-person legislation, the party leader added.
Among the party’s major gains was Ms Davidson winning the Edinburgh Central constituency from the SNP and deputy leader Jackson Carlaw taking Eastwood on the outskirts of Glasgow from Labour’s Ken Macintosh.
Two Tory MSPs were elected in Glasgow, where academic Adam Tomkins and food retail manager Annie Wells picked up seats.
The SNP’s Dennis Robertson lost his Aberdeenshire West seat to the Conservatives while Oliver Mundell – son of the Scottish Secretary – captured the Dumfriesshire constituency from Labour.
The Tories’ previous best result at Holyrood was 18 MSPs, a total the party achieved in both 1999 and 2003.
Ms Davidson congratulated Ms Sturgeon on the SNP victory but said the lack of a majority government was “very welcome and good for our democracy”.
She said: “Majority government has not worked well – too often over the last five years the SNP pushed through its agenda not on the strength of its case, but simply on strength of numbers.
“As a minority administration, I believe the SNP will be forced to listen, to learn and to improve.
“I am very very proud that our performance last night has helped to bring this about.”
Asked if she thought the decline of Labour in Scotland was “terminal”, she said: “I believe in politics that the wheel always turns and I think that means you can get past peak Nat just as you can get past the Labour trough.
“So, I don’t believe that anything in Scotland is permanent.”
Ms Davidson insisted her party would not “moan from the sidelines” as she claimed Labour has done over the past five years in the Scottish Parliament.
“Where we need to oppose, where we need to challenge, where we need to block, then we will do that too.”
Mr Cameron congratulated his Scottish leader in a message on Twitter.
He added: “She is a leader who will stand up to the SNP and give Scotland strong opposition.”
Scottish Secretary David Mundell said it would be “good for Scotland, good for the Scottish Parliament” to have the Tories as the official party of opposition.
The Conservative MP said: “Ruth has demonstrated throughout this campaign that she is the one politician that can go toe-to-toe with Nicola Sturgeon, that she can stand up to demands for a second referendum.
“She can also scrutinise the SNP and make sure they deliver on promises they have made on health, on education, and I think by doing that we will get better government and that will be good for everyone in Scotland.”
He added: “It will be a seismic change in Scottish politics that the Scottish Conservatives are the second party in the Scottish Parliament.
“I was a candidate back in those first elections in 1999, it would have been incredible to think the Scottish Conservatives could have finished ahead of Labour and be the official opposition. It demonstrates that Ruth has transformed our party in Scotland, taken it forward.”