The 32-year-old was out of contract in the summer, but has now pledged his future until the end of the 2023/24 season.
Rodriguez is joint-top-scorer this season with seven goals in all competitions – alongisde Maxwel Cornet – with the highlights being superbly-taken equalisers against Manchester United and Everton at Turf Moor, as well as four goals in 26 minutes against Rochdale in the Carabao Cup.
Rodriguez started his career with the Clarets,before becoming the club’s then-record sale in 2012, joining Southampton for £7m, before returning in 2019 from West Brom, and he added: “The club means a lot to me. Having grown up in the area, it is always a pleasure and a privilege to represent Burnley, so when the club offered the contract, it was a no brainer for me. I couldn’t wait to sign it.
“To play for Burnley being a Burnley lad is extra special. To see where the club has come and where it’s at now, I just wanted to be a part of that and I can’t wait for the next chapter at the club.”