After making stunning gains in the 2015 election, Nicola Sturgeon’s party lost seats to the Conservatives, Labour and the Liberal Democrats.
Ruth Davidson, whose Scottish Conservatives ejected Mr Salmond from Gordon and Mr Robertson from Moray, said her party is enjoying an “historic night”.
She told BBC Scotland: “Indyref2 is dead, that’s what we have seen tonight.”
Ms Sturgeon, who succeeded Mr Salmond as SNP leader and First Minister in 2014, admitted she was “disappointed” by the results, but said she would not make any “rash decisions” on her plan for another independence vote.
The Tory success in Scotland is in stark contrast to the situation south of the border, with Theresa May facing the prospect of losing her majority in the Commons.
The First Minister told BBC Scotland: ‘’This has been a disaster for Theresa May. She called an election clearly very arrogantly thinking she was going to crush the opposition, sweep everybody aside and cruise to a landslide majority.
‘’Her position I think is very, very difficult. We have to wait and see how things shake out. I’ve always said the SNP would want to be part of a progressive alternative to a Tory government.
‘’I’m disappointed at the SNP losses but I’m pleased that we’ve won the election.’’
That was echoed by Mr Salmond, who said the SNP would seek to “build a progressive alliance to take this country forward and to avoid the calamity of hard Brexit”.
The former nationalist leader, who was first elected as an MP 30 years ago in 1987, was defeated by Tory Colin Clark.
In his victory speech, Mr Clark said: “The silent majority have spoken. We’re proud to be part of the United Kingdom.”
Mr Salmond blamed a late surge in support for Jeremy Corbyn’s Labour for the SNP’s defeat in many seats, with the nationalists also losing John Nicolson, Mike Weir and Tasmina Ahmed-Sheikh.
Pete Wishart, who was the chair of the Scottish Affairs Committee in the last parliament, held on to his Perth and North Perthshire seat by just 21 votes, with Tory MEP Ian Duncan coming close to ousting him.
Labour gained a number of seats back from the SNP - with the party making a return to winning ways in Scotland’s largest city, taking the seat of Glasgow North East from Anne McLaughlin.
Ian Murray, who had been the only Labour MP in 2015, retained his Edinburgh South seat with a massively increased majority.
The Liberal Democrats also enjoyed success in East Dunbartonshire, where former UK government minister Jo Swinson won back the seat she had lost in 2015.
Mr Salmond said: “The Scottish National Party have lost many fine parliamentarians this evening, and that is a grievous blow to the SNP.
“But overall the results in Scotland show the SNP will have won a majority of seats in this country and a majority of the vote - something which I suspect the Prime Minister would like to be able to claim in the early hours of this morning but may not be able to do so.
“So the SNP might well find itself in reduced numbers in the House of Commons, but in a position of very substantial influence indeed.
“And I know that my colleagues will seek to use that influence to keep the Conservative Party from power and to build a progressive alliance to take this country forward and to avoid the calamity of hard Brexit.”
Scottish Labour leader Kezia Dugdale said her party is seeing some “really encouraging results”.
She told BBC Scotland: “Jeremy Corbyn’s anti-austerity message has focused on radically changing our country and who it works for, so it actually delivers right across this country.
“The SNP vote is crumbling in their heartlands. Look at what’s happening in Glasgow; look at what’s happening across the west. It’s a very bad night for the SNP.”