Clarets striker Rodriguez swivelled and fired home from inside the box following a James Tarkowski header to reward Sean Dyche’s side for their resilience and persistence in the face of large periods of home dominance.
The visitors could even have left Anfield with all three points, but Johann Berg Gudmundsson fired an effort against the crossbar with just minutes remaining.
Andy Robertson had given the home side the lead in the first half, but they also spurned a number of good opportunities to score more, with Mohamed Salah and Curtis Jones guilty of wayward shooting.
They also found Nick Pope in superb form, with the England goalkeeper producing a string of good saves to deny Salah and others.
Roberto Firmino struck the inside of the post with a low shot as Liverpool poured forward to try to capitalise on having more than 70 percent possession.
The draw means Liverpool can now no longer beat the Premier League record for home wins in a season; they need to beat Chelsea in their last game at Anfield to equal the league-high of 18.
Liverpool remain in good shape, though, to achieve a highest-ever Premier League points haul; with three games left, they have 93, seven short of the record of 100 set by Manchester City in 2017-18.
Burnley are now five games unbeaten since they returned from the Premier League’s three-month shutdown by losing 5-0 at Manchester City.
A draw leaves them ninth and still firmly in the mix for a place in next season’s Europa League.