The defensive midfielder picked up a yellow card for a foul on Lichtsteiner during the first-half of Serbia's 2-1 World Cup defeat to Switzerland, but that left no lasting mark on the right-back's opinion of the 29-year-old.
Lichtsteiner will come up against Matic next season in the Premier League as Arsenal begin life under new manager Unai Emery. The Swiss captain is one of many new arrivals at the club, joining goalkeeper Bernd Leno and defender Sokratis Papastathopoulos, who is due to arrive in London at the start of July.
The 35-year-old's first English top flight game against United will come on December 4 when Arsenal visit Old Trafford and Lichtsteiner is well aware of the qualities Matic can show for their rivals.
"Matic is an amazing player. Also as a player, I like him a lot," Lichtsteiner told Goal.
"I have heard from many people that he his a nice person. He has amazing qualities and is an amazing character."
The new Arsenal signing also had praise for former Premier League players Xherdan Shaqiri and Aleksandar Kolarov, praising his international team-mate for his impact against Serbia where he scored the late goal to seal the win.
Kolarov was part of a determined Serbia side who took the lead before goals from Albanian-Swiss duo Granit Xhaka and Shaqiri sealed three points for the central Europeans, with the pair both celebrating in controversial circumstances.
"I am very happy with Shaqiri. Very happy. He played very well, as did all our players.
"Kolarov is also very good. I like the Serbian team. But there is this history between them and the Albanians in our team, so it was not easy."