The forward doubled his tally three minutes later with a sharp finish to give Celtic a stranglehold on the match, before Stuart Armstrong and Roberts wrapped up victory with second-half efforts.
SPL top scorer Sinclair completed his hat-trick with a penalty in the 84th minute as Celtic moved up to 86 points after 30 matches, 25 points clear of second-placed Aberdeen, and 35 ahead of Glasgow rivals Rangers in third.
"I've grown up with the club - and to be responsible to give the supporters that joy is really humbling for me," Rodgers told Celtic's official website. (www.celticfc.net)
"We never get tired of trying to get better, and of trying to improve," he continued. "That's something we've tried to impose onto the players and their mentality, and we always have to look to be better."
Rodgers' side are still unbeaten having drawn just two matches and having already won the League Cup. They have a Scottish Cup semi-final against Rangers coming up on April 23 so remain on course for the treble.
"To win the title without losing a game is, of course, a great achievement," added Rodgers.
(Reporting by Ed Dove; editing by Clare Lovell)
