Douglas Ross, who has had a difficult relationship with Downing Street in recent months, said Ms Truss had “shown herself to have a record of delivery”.
Meanwhile, Nicola Sturgeon challenged Ms Truss to act immediately on the cost-of-living crisis.
Their relationship became a key talking point in the Tory leadership contest after Ms Truss accused Ms Sturgeon of being an “attention-seeker” who was best ignored.
Ms Truss has promised to deliver a “bold plan” to cut taxes and grow the economy, as well as immediate action on energy bills.
It comes after an often bad-tempered leadership contest in which she clashed with Mr Sunak in heated TV debates.
Mr Ross, who did not publicly back either candidate, will be hoping Ms Truss’s election paves the way for a fresh start.
He called for Boris Johnson to resign in January following the Downing Street ‘Partygate’ revelations.
Last month, Mr Ross told an audience at the Edinburgh Festival Fringe that Mr Johnson was angered by the move and used “rude words”.
The Scottish Tory leader later U-turned on the position, but voted against Mr Johnson in a confidence vote in June.
In a statement, he said: “Congratulations to Liz Truss on her election as Conservative Party leader.
“Both Liz and Rishi, along with their teams, should be proud of the campaigns they have run and the way they have engaged party members.
“However, now that the leadership election is over, it is time for the whole of the Conservative Party to come together to tackle the big challenges our country faces.
“Throughout her political career, Liz Truss has shown herself to have a record of delivery, whether that was in removing the US tariffs on whisky and cashmere or in managing the UK’s robust support for Ukraine both before and following the Russian invasion.
“I look forward to working with our new leader and Prime Minister as she continues to deliver for Scotland and the whole of the UK at this most challenging of times.”
SNP Westminster leader Ian Blackford insisted Ms Truss must now "get on with it" and deliver a major package of cost-of-living support.
He said: "All the signs suggest Liz Truss is shaping up to be even worse than Boris Johnson, with the Tories lurching further to the right, and continuing to impose damaging policies like the extreme Brexit that has raised the cost of living.”
Scottish Labour leader Anas Sarwar said the announcement was “not a fresh start for the Tory party or for our country”.
He said: “Like many people, I breathed a sigh of relief when Boris Johnson resigned. But my relief at his departure was short-lived.
“Liz Truss might claim to be different to her predecessor, but the grim reality is that a Tory party that has lost touch with reality will still be failing families across the UK.
“Instead of the fresh start our country needs, the best the Tories can deliver is more deceit, division and distraction.
“All the while, families worry about how they will afford spiralling bills and sky-high energy costs.
“The cost-of-living crisis is the number one issue facing families today, and the Tories have no answers.”
Scottish Liberal Democrat leader Alex Cole-Hamilton said the Tories had shown “they don’t care, have no plan and have failed our country”.
He said: “Alongside the SNP, the Conservatives are part of the clash of nationalisms that has held Scotland back.
"The Government needs to scrap October’s energy price rise to avoid a social catastrophe for families and pensioners this winter.
"We need a general election, to get the Conservatives out of power and deliver the real change the country needs."
Scottish Greens co-leader Lorna Slater called Ms Truss’s election “terrible news for Scotland and our environment”.
She said: “Liz Truss has supported every single one of Boris Johnson's most damaging decisions.
"Whether it was the Brexit deal that has curbed our right to travel while driving up prices, the cruel Universal Credit cut that has taken vital support away from thousands of the most vulnerable people and families, or his failure to act during the worst cost-of-living crisis for decades.
“Her proposals for a major expansion of North Sea oil drilling would leave us even more dependent on fossil fuels while doing real and lasting damage to our environment at a time when we should be investing in the renewable jobs of the future.”
Elsewhere, the Scottish Tourism Alliance requested emergency support from the new administration.
Chief executive Marc Crothall congratulated Ms Truss on her appointment, adding: “Our asks of the new UK administration are clear.
"We need an urgent freeze on the energy price cap for households and businesses, VAT reduction on supplies relating to hospitality, accommodation and attractions, deferral of the Coronavirus Business Interruption Loan Scheme repayments, urgent support for businesses struggling to secure new energy contracts and an immediate review of immigration rules to address the labour workforce shortages within the sector.
“Securing a total waiver of business rates for all tourism and hospitality rateable businesses with immediate effect from October 1 for a minimum of 12 months must be a priority in the raft of emergency responses to the cost-of-living crisis, which the Scottish Government will announce over the coming weeks. It is imperative that Westminster provides the necessary funding to the Scottish Government to enable this critically important measure to be implemented.”