Leigh Griffiths gave the champions the lead after 34 minutes at Rugby Park in comical style when Greg Taylor's scuffed clearance cannoned off the bar and fell straight to the striker.
But, after Chris Burke's equaliser half an hour later, Celtic were undone by a stoppage-time winner from Stuart Findlay.
The result leaves the Bhoys sixth in the table, six points behind league leaders Hearts, who beat Rodgers' men 1-0 earlier in the season.
"It hasn't been a great start to the campaign but we have to accept that," Rodgers told a post-match news conference.
"We're the only ones who can make it better by playing better and getting more consistent results.
"It's been very bitty but we have to go away and work and see where we can be better."
Celtic meet St Johnstone in the League Cup on Wednesday before entertaining Aberdeen at Celtic Park on Saturday, and Rodgers believes fatigue from Thursday night's 1-0 victory over Rosenborg in the Europa League may have been a factor.
"I take responsibility," he added.
"We made the changes to try and get that energy and brightness and sharpness that may not be there if you've played late on a Thursday night.
"I'd have been disappointed to draw the game but to actually lose is bitterly disappointing."
The Hoops' woes were further compounded by rivals Rangers romping to a 5-1 win over St Johnstone on Sunday, which moved Steven Gerrard's side up to second, one point ahead of Celtic.