Matic started and completed 45 minutes as goals from Henrikh Mkhitaryan and Juan Mata secured victory in the friendly at the Aviva Stadium in Dublin.
"I trained alone in London and it was strange for me but I tried to give my best to control the game well and to give balance," Matic told MUTV .
"I think for the first game it was good for me.
"Step-by-step I think I will be better and better and I hope I will help the team to achieve something this season."
Matic's composed debut will have enhanced his chances of appearing in a competitive match for the first time in Tuesday's UEFA Super Cup encounter with Real Madrid.
Despite his limited preparation, United's newest signing is confident of being fit to play against the reigning Champions League winners.
"I am sure I will be 100 per cent ready and we can do something good with this club," he said.