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.
Andrew 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% 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.
They remain in decent shape to achieve a highest-ever Premier League points haul; although with three games left and seven points short of the record of 100 set by Manchester City in 2017-18, they have no room for error.
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.
Liverpool's main goal this season has already been achieved, with the title taken from Manchester City in emphatic fashion.
Jurgen Klopp may have said this week that surpassing City's Premier League points record of 2017-18 was "not that important" to him, but it would provide a fitting cap to a stunning campaign.
Three wins from their remaining three games will see Klopp's side set a new benchmark, but it is not the only record still possible for them this season.
In reaching 100 points, City also set a record Premier League title-winning margin - finishing 19 points clear of their nearest challengers. Liverpool are 26 points ahead of City, albeit having played a game more.
They are also still on course to claim the most wins in a Premier League season (currently 32) and most home points in a campaign (55).
This was a small reminder, though, that for all the strides they have made over the past few seasons, the Reds remain fallible.
A bit of luck would have seen Liverpool home, with a post denying Firmino, while Pope was inspired, but Klopp's side also contributed to their own failure to win.
Salah was extremely wasteful, while Jones twice fired wide during an otherwise impressive display as midfield deputy for the injured Jordan Henderson.
At the other end, Alisson and Virgil van Dijk - for all their ability - can be as vulnerable as any to a well-placed delivery and a physical, aerial threat.
Disappointment then for the Reds, but a timely reminder that they are there to be got at and never truly the finished article.
Burnley continue to make a lot out of little
Burnley's winless run at Anfield was extended to 13 matches by this draw, but they will only take positives from this display, offering further proof they are one of the Premier League's toughest nuts to crack.
Wave after wave of Liverpool attacks came and went without a goal as defenders Tarkowski and Kevin Long stood tall while goalkeeper Pope produced heroics.
They had just 29% possession, and only six shots to their opponents' 23, but Dyche's side are one of the masters of making a little go a long way.
Since losing at City on 22 June - their only defeat in 13 league games - Burnley have picked up 11 points from five matches, conceding just twice.
And all this has been achieved while dealing with an injury and contract crisis that deprived them of a host of their first-team regulars.
There is now light at the end of the tunnel in that regard, with Chris Wood fit to partner Rodriguez in attack again and Gudmundsson further stepping up his return.
It could be just the boost they need to push them over the line for another impressive European qualification.
Man of the match - Nick Pope (Burnley)
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