The Scotland Under-21 international has agreed a three-year deal, the Barclays Premier League newcomers have confirmed.
Reid, speaking at New Douglas Park as Accies announced a new one-year, six-figure shirt sponsorship deal with Reid Furniture, believes the former Hamilton youth player deserves his big move.
"I am not going to say I'm disappointed," the Accies boss said. "It's a dream for him, an opportunity for a young man from Kings Park to play in the Premiership.
"It's deserved and everybody at the club is delighted for him. We can replace Brian on the park, we have James Gibson who can come in. But Brian was a fantastic character, every morning he came in to his work with a smile on his face and that's what I will miss the most."
Reid is confident that Easton can take the next step up in the international ladder if he pins down a first-team place at the newly-promoted Lancashire club.
He said: "Brian is a fantastic player. I watched him in the B international for Scotland and he will go a long way in the game. I'm hopeful he can make an impression at Burnley.
"The important thing is to get in the team and play consistently and if he is a Scottish player playing in the Premiership consistently, then I think that will enhance his reputation and give him a better chance of a full international cap."