The British number one was disappointing in Davis Cup defeats by Dutchman Botic Van De Zandschulp and Kazakhstan's Alexander Bublik and then lost a close match to Taylor Fritz at the Laver Cup last weekend.
But he took advantage of a kind draw in Seoul with a 6-2 6-2 victory over Japanese wild card Uchida, ranked 163.
Norrie, who as the second seed had a first-round bye, served eight aces and did not face a break point as he set up a quarter-final against American Jenson Brooksby.
"It feels great," said Norrie. "I was a bit jet-lagged, so it was nice to get up early and get it done in straight sets.
"I felt the conditions suited me well, it was pretty hot and I like that. It was a tough match but I was able to win some close games and get a couple of breaks each set so it was a good day for me overall."
World number two and US Open finalist Casper Ruud also received a bye into the second round and will be in action on Thursday against Chilean Nicolas Jarry.