Ruth Davidson has said she will argue Scotland's case "forcefully" to the prime minister after major Tory gains north of the border.
The Scottish Conservative leader said she wanted to build consensus around a Brexit deal that put economic advancement at its heart.
She also dismissed reports that Scots Tories might "break away" from the main UK party.
The Scottish Conservatives won an extra 12 seats in Thursday's election.
The performance of Scottish Tories was in marked contrast to that in England and Wales where Prime Minister Theresa May lost her majority in the House of Commons.
Ms Davidson spoke to reporters in Stirling as she showed off the 13 Scottish Conservatives who will now take their seats at Westminster.
She said: "I'm going to make sure that along with the 13 MPs we're sending to Westminster, we put Scotland's case forcefully and make sure that we deliver for Scotland and make sure that the union delivers for Scotland too."
She added: "I want to ensure that we can look again at issues like Brexit which we know we are now going to have to get cross-party support for - and move to a consensus within the country about what it means and what we seek to achieve as we leave."
Asked what she meant with her call for an "open Brexit" she said: "I've never believed in the terms of hard or soft Brexit.
"I want to talk about open or closed - and as someone who believes in open politics, that's about making sure we tear down barriers rather than put them up.
"It's about making sure we put free trade and economic advancement at the heart of the Brexit deal as we leave."
She also said she had received assurances from Theresa May that any deal with the Democratic Unionist Party would not erode LGBTI rights in Northern Ireland, and that the UK government would use its influence to "advance" those rights.
Earlier she had taken to Twitter to dismiss a report in the Daily Telegraph that she wanted to "tear her Scottish party away from English control" .
The paper, which stands by its report, said there were tensions during the campaign, and that Ms Davidson's aides were working on a deal to set up a separate organisation in Scotland.
Pressed on the issue in Stirling, she said: it had been decided when she became leader that "we would have complete autonomy over policy, candidates, campaigning, finance, which is what we have... but we would take the Conservative whip at Westminster and that is exactly what is going to happen".
Ms Davidson also confirmed she had given her "full support" to Mrs May staying on as prime minister.
Earlier the SNP's cabinet secretary for finance and the constitution, Derek Mackay, insisted his party had won the election in Scotland, despite seeing its number of MPs fall from 56 to 35.
He told the BBC's Good Morning Scotland programme: "The 2015 result was incredible, very difficult to repeat - but we won more seats than all the other parties put together."
He said it was too early to say whether the SNP would now "park" its demand for a second independence referendum after 60% of Scottish voters backed pro-union parties.
"Of course we will listen and reflect. It's too premature to say what we would do next around that," he said.
"We will continue to deliver good governance for Scotland, stand up for Scotland in the Westminster parliament, make sure we can get the best possible deal for Scotland in terms of the negotiations on Brexit."
Scottish Labour demanded Nicola Sturgeon ditch plans for a second independence referendum.
James Kelly, who was the party's election campaign manager, said: "Nicola Sturgeon has some very serious questions to answer now.
"She must categorically drop her plans for a divisive second independence referendum and get back to the day job.
"Across the UK, the election showed there is huge support for Labour's vision for investment in our public services, and Nicola Sturgeon needs to accept that she has been too distracted by the constitution for too long."