It was their fourth victory in 21 Tests in England over 40 years and their first since a 1-0 victory in a two-Test series 10 years ago
Back-to-back 1-1 draws against Leicester and West Ham enabled City to edge ahead on goal difference before Liverpool reclaimed top spot with a 3-0 win over Bournemouth on Saturday as they chase a first top-flight title since 1990.
But, Robertson said, the German manager is unflustered, despite seeing their lead disappear.
"He has been calm and the backroom staff have been a different class," said the Scotland captain. "They are the team behind the team and they are the ones who keep us going and positive.
"When people say he is nervous or we are nervous it couldn't be further from the truth. He's probably been more 'smiley'. He is just trying to get us going and it's obviously worked.
"In terms of the players, we've not seen any difference in any of the staff and that feeds into the players and we've not changed. Whether we draw, win or lose it doesn't change."
The performance against Bournemouth, who have now lost nine away matches in succession and have suffered 13 defeats in the past 18 matches in all competitions, showed a return to the high-energy pressing game and clinical counter-attacking that has been so successful for Liverpool.
Sadio Mane scored in a fourth successive Premier League match for the first time and there were also goals for Georginio Wijnaldum and Mohamed Salah.
Liverpool fly to Marbella on Monday for warm-weather training with their season firmly back on track.
"I don't think it's a message to any other team or anything," said 24-year-old Robertson. "It's maybe a message to our fans and everyone connected to the club that we're here.
"We fought for everything and got the result that we needed."
It was their fourth victory in 21 Tests in England over 40 years and their first since a 1-0 victory in a two-Test series 10 years ago
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