The 20-year-old starred for the Clarets at right- back in an important 2-1 win over bottom club Charlton Athletic 24 hours after agreeing to the loan deal at Turf Moor.
Of Williams' first game, Coyle said: "I thought it was an absolutely magnificent debut from somebody who had just turned 20 years of age, with the athleticism he showed and being comfortable on the ball.
"Rhys only trained with the boys on Friday for an hour or so, but it was a magnificent debut.
"For me, he put the cross of the match in during the first half that should have resulted in a goal.
"He'll go on to be a top, top player, and hopefully we can help him here as much as he can help us.
"He's one with a big future. I tried to get him in the summer, and he falls into the blueprint of what we're trying to do at the football club, with young, athletic, energetic players coming in to complement the ones that we have and I hope the rest of his games are as good as that.
"I really believe he'll go back to Middlesbrough a far better player for the experience of playing with us, and I'm sure he can help us in the meantime."