The Burnley-born England Under-21 striker opened the scoring in clinical fashion in the 22nd minute and then embarked on a delightful run to provide a second for Wade Elliott a minute later.
Eight minutes before the break, defender Michael Duff was allowed the freedom of Turf Moor to head home unopposed from a corner as Eddie Howe's men moved to within striking distance of sixth-placed Leeds.
The Clarets face Simon Grayson's side at Elland Road on April 30, and having ended a dismal six-match winless sequence with victory over Swansea on Saturday, they may feel they are beginning to rediscover their best form when it matters most.
Middlesbrough replied through Andrew Taylor's late strike but the defensive flaws which have undermined them all season were glaringly apparent during the first half.
The 2-1 weekend win over Swansea had evidently poured confidence into Burnley and they cut the Boro defence to shreds during the opening 45 minutes.
In the 17th minute, Chris Eagles crossed dangerously for Rodriguez to head narrowly over the crossbar.
Four minutes later, Eagles collected possession 20 yards from goal and hit a low right-foot shot which Boro goalkeeper Paul Smith was forced to smother.
Less than a minute later, however, Burnley led when Danny Fox's low inswinging corner from the right flank was dummied by Eagles for Rodriguez to drill home a low right-foot shot from 10 yards.
It was the 21-year-old's 13th goal of a fine personal campaign and he further enhanced his cult status with Burnley supporters a minute later.
Collecting possession just inside Middlesbrough's half, Rodriguez combined searing pace and clever footwork to glide past a couple of visiting defenders and find Elliott lurking on the right-hand side of the 18-yard box.
The midfielder received Rodriguez's low pass and thumped a fierce right-foot shot into the net to give Smith no chance.
Boro conceded again in the 37th minute when Fox delivered another inswinging corner from the right flank and the unmarked Duff planted a firm header into the net at the far post.
Five minutes after the break some brilliant footwork from Eagles almost led to a fourth Burnley goal and then Rodriguez curled a fine shot just over.
Boro substitute Leroy Lita almost made an instant impression with a fierce shot on the hour but the visitors did claim a consolation in the minute.
Former Burnley loanee Rhys Williams' sweet strike from 18 yards rebounded off the inside of a post and was swept home left-footed by Taylor.