The Japanese player had knocked out the Scottish former World No 1 in straight sets in the Australian Open in January.
"He played very well in Australia, he had a very good run there and was too good for me there," Murray said after his victory on Tuesday.
"I tried to be the one dictating from the first point, and I thought I did that well. It was one of the better matches I've played in recent months."
Murray broke Daniel twice in the first set.
Daniel broke in the first game of the second set, but Murray broke back at once. He broke again two games later and in the eighth game to clinch victory in one hour, 20 minutes.
Murray, 34, has won the Qatar title twice: in 2008 and 2009.
"Obviously the results from 12 years ago aren't going to affect the results this week," he said.
"But what it tells me is that the conditions here are good for my game, so if I can play to a good level the courts are going to suit me here and I'll make it difficult for everyone I play against."
Murray will face second seed Roberto Bautista Agut in the second round.
Murray has won three of his four meetings with Bautista Agut, but all those victories came before 2016 and the Spaniard won their last encounter in 2019.