Andrew Robertson headed Liverpool into a 34th-minute lead, brilliantly sending Fabinho's cross beyond Pope.
In first-half stoppage time, Mohamed Salah's touch set up Sadio Mane but his shot was superbly saved by Pope.
Roberto Firmino almost had his first home Premier League goal this season but his strike four minutes into the second half hit the post.
Burnley were level on 69 minutes after Rodriguez brought down James Tarkowski's headed pass, before turning and firing into the bottom left corner.
Burnley substitute Johann Gudmundsson struck the crossbar before Pope again denied Salah as the Clarets halted Liverpool's 24-match winning sequence at home.
The Reds move on to 93 points, while Burnley are up to ninth place with 50.
Liverpool: 15 July v Arsenal (A)
Burnley: 15 July v Wolves (H)