The in-form Blues threatened to leave Turf Moor with a repeat of their 4-0 win of last season.
Sean Dyche's men were also staring at the prospect of a heaviest home defeat since Boxing Day after a hat-trick from Christian Pulisic. But the Clarets hit back with a spectacular strike from Rodriguez and Dwight McNeil's deflected effort in a frantic final five minutes to finish on a high.
“Maybe on another day we take the chances at the right time and it becomes a different game," admitted Rodriguez, who started in attack alongside Ashley Barnes for the first time after injury ruled out Chris Wood.
“We took the game to Chelsea at times. Against these teams you've got to take your chances and punish them like they punished us. That's why they are a top side.
“We felt we were still in the game at 2-0 but unfortunately, we conceded two more and against a team like that is always difficult.
“But the belief we still had to keep working and go and get two goals and have that hope that we could have got something was real credit to the lads.
“It's all ifs and buts, but what we can take into the next game is the belief that we can hurt top teams.
“We showed character and that's what this group of lads is about. There's not one of them will shy away from anything and that's something to be proud to be a part of.
“They are full of character and won't give up until the end and at one point you thought we might get something from it. That shows what kind of lads we've got in there."