Robertson was praised for his leadership qualities and role within the Scots' dressing room after their victory over Armenia.
Clarke's side eased their World Cup blues with the perfect start to their Nations League campaign, securing a 2-0 win at Hampden Park in their Nations League Group B1 opener, thanks to headed goals from defensive duo Anthony Ralston and Scott McKenna.
Robertson found himself the target of trolls at the end of what as an incredible season at club level, with rival fans taking aim at the FA Cup and Carabao Cup winner after Liverpool's Premier League title near-miss and Champions League final defeat.
And then again after Scotland missed out on a place at the 2022 World Cup in Qatar.
The 28-year-old admitted the end to the season was 'the toughest ten days of my career' as he reacted to those three huge misses.
"Personally, it's been the toughest 10 days of my football career, of course it has been," said the Liverpool defender after Scotland's World Cup play-off defeat to Ukraine.
"Emotionally, mentally, physically, everything. So it's not been great.
"But to be honest I will deal with that myself, I will go away and think about that myself.
"But I am just gutted for those lads in there. Obviously I was desperate for them to get to a World Cup and play on the biggest stage and unfortunately we have fallen short.
"But we need to be ready the next time it comes around. The way it is with international football, people get older and you never know when your last chance of it will be. That's why it hurts so much."
But Clarke insists the left-back deserved cheers, not jeers - saying Robertson has played a true captain's part on and off the park for his country.
"I have to mention the captain, I thought he was outstanding," said the Scotland boss.
"He was outstanding in the dressing room before the game, himself, John McGinn, Callum McGregor, leaders, taking us on the pitch in a good frame of mind and with three good performances.
"But I thought the captain was exceptional. It was just in general, as a captain, leader."
"It is a role that Andy has really grown into and sometimes he doesn't quite get the mentions that maybe he should do.
"And captains are important at this time, people you are looking for to step up, John McGinn, Callum McGregor that was important as well."