Born and raised in the Lancashire town, the 24-year-old spent five seasons at Turf Moor before making the move south in June 2012.
It was the first time the forward had faced his former club and declined to celebrate his goal in the 4-3 victory, which saw both sets of fans greeting his 35th-minute introduction with a warm ovation.
"I did it out of respect to my old club," Rodriguez said of his refusal to celebrate. "I still support them and still look out for them. I think it was good to show respect to them.
"Obviously we are disappointed with the goals conceded, but we are through to the next round, which is the man thing.
"It is very important and every player in there wants to go as far as we can in the FA Cup.
"It is a big cup for us and we want to get a good cup run going, so it is very important for us to make it through each round."
Adam Lallana's long-range effort meant a late Kevin Long goal counted for nothing, but the Clarets' fight makes Rodriguez confident of facing them in the top-flight next term.
"They made it tough," Rodriguez said. "They played really well, created some chances and put them away.
"I think Ingsy and Vokes are on fire at the moment, so there is no surprise there.
"You have always got to be wary of Burnley. They have scored a lot of goals and they showed that again. I thought we did really well to get the win."
Saints will hope the victory spurs them onto better things in the Barclays Premier League, having won just once in their last nine matches.
Next up is a match against West Brom and, with club-record signing Dani Osvaldo suspended and Lambert forced off against Burnley with a dead leg, top scorer Rodriguez may well have to add to his 10-goal tally for the season if Mauricio Pochettino's side are to win.
"I have been a little disappointed with the chances I have missed - that is how I look at it," Rodriguez said.
"I want to try and improve and improve this tally. I want to keep working at it and try and make every chance count. Hopefully I can carry on if I keep working hard."
- Sports & Recreation
- Rickie Lambert