Callum McGregor fired the champions into an early lead in Paisley and Reo Hatate doubled the visitors' lead before the interval.
Alistair Johnston netted the third in the second half to underline Celtic's superiority.
The Hoops have won all four games in all competitions this season, including three in the league, scoring 12 and conceding just one in the process.
They go into next Sunday's Old Firm game against Rangers at Parkhead once again looking firm favourites to win the title.
But Rodgers confirmed the absent Matt O'Riley is on his way to join Premier League club Brighton after a quiet transfer window for the Glasgow giants.
Goalkeepers Kasper Schmeichel and Viljami Sinisalo have joined, while striker Adam Idah and midfielder Paulo Bernardo have returned following loan spells.
Rodgers, who hopes at least some of the O'Riley transfer fee will be used to strengthen the squad, believes the Celtic board owes it to the supporters to build on their current success.
"There are a number of things I won't go into here. But it's definitely something, having come back in and gone through three windows now, we have to put right as a football club," Rodgers said.
"We shouldn't have been getting into this last week in the position we're in. That's the reality. However, that's something for us for after this window.
"As I said, we'll get the players in that we want. It's taken a lot longer than I would have liked as the football manager. That's my brutally honest answer on it."
There were less than three minutes played when McGregor picked up a pass from James Forrest outside the box and drilled a shot low past Buddies keeper Ellery Balcombe for his second goal of the season.
There was a VAR check for a Celtic penalty on the half-hour mark when Daizen Maeda went down in the area following a challenge by Alex Gogic, but the decision went against the visitors.
Celtic increased their lead anyway as Johnston's lofted cross to the far post was headed back by Maeda for Hatate to lash a shot low in off the far post.
Kyogo Furuhashi and Nicolas Kuhn replaced Idah and Forrest just before the hour mark.
Balcombe thwarted Furuhashi after the Japan striker raced on to Johnston's defence-splitting pass.
But there was no halting Celtic's momentum and in the 71st minute, Kuhn drifted into the St Mirren box and set up Johnston to knock the ball high into the net from 10 yards.
There was also time for 18-year-old Francis Turley to make a late debut for Hatate as Celtic, who showed they remain a potent force even without the services of influential Denmark midfielder O'Riley.
