Peterborough were without a win for four games in all competitions, but caught Burnley cold in freezing temperatures in the first half.
A defensive lapse allowed Paul Taylor to give Posh a 12th-minute lead and David Ball rattled the post as the visitors posed plenty of problems in the opening stages.
However, after the break Burnley penned their opponents back with little reward as chances went begging.
The Clarets had a strong penalty appeal rejected in the closing minutes when Ross Wallace's shot was blocked by Mark Little's outstretched arms.
But with three minutes remaining Rodriguez set off on an incredible solo run, evading four challenges before coolly slotting home for his 17th goal of the season and fifth in his last five matches.
Little and Ball came into Peterborough's defence and attack respectively, while Wallace returned for Burnley as Keith Treacy dropped to the bench in the only change from the XI that beat Nottingham Forest on Tuesday night.
An early chance fell to Clarets striker Charlie Austin, but Taylor was back inside his own area block the goalbound effort.
Taylor was soon reaping the benefits of his early intervention when he capitalised on an error from Burnley defender Ben Mee and smashed past Lee Grant from 12 yards to give Peterborough the lead.
Moments later the home side were caught napping at the back once more, sending Ball clear to plant a shot against the inside of the post before a grateful Grant smothered the rebound with George Boyd in close attendance.
At the other end Wallace looked to fashion a response with a vicious drive that bounced awkwardly and forced Posh goalkeeper Joe Lewis to parry clear.
But defensive frailties were still in evidence for Eddie Howe's side and Boyd should have done better when he drilled wide with only Grant to beat in the 21st minute.
Burnley were struggling to make an impression on the Peterborough back line, but with 37 minutes gone Rodriguez released Kieran Trippier down the right and Austin's header from the full-back's cross drew a superb point-blank save from Lewis.
The Clarets began the second half with renewed intensity amid increasing snowfall and captain Chris McCann had a glorious change to equalise, but his free header from Trippier's free-kick after the hour went over.
Rodriguez then carried the fight to Peterborough with a driving run and shot that forced another sharp Lewis stop.
From the resulting corner, Rodriguez's shot was blocked and he went down under a challenge from Posh defender Gabriel Zakuani when attempting a follow-up, although referee Colin Webster remained unmoved.
Austin had a 69th-minute effort from a Josh McQuoid cross headed off the line as the Peterborough goal continued to lead a charmed life.
McQuoid soon made way for Treacy who immediately prompted some slick interplay with Wallace to slash across goal from inside the box.
The hosts were left seething as Webster waved away claims for handball against Little when Wallace unleashed a shot in the 84th minute, but Rodriguez soon had the home support in raptures with a truly special strike.