A Matty Fryatt double looked to be steering Hull to a comfortable Championship victory at the KC Stadium on Saturday, before David Edgar set up a tense finale with two goals in four minutes.
And Rodriguez's 10th goal of the season in the third minute of stoppage time gave Burnley a much needed victory and left the home fans shell-shocked.
Chris McCann almost put Rodriguez through on goal in the opening minutes but Jack Hobbs' saving tackle prevented the him from getting a shot away.
Junior Stanislas twice had opportunities to put Burnley ahead on his 22nd birthday, but wasted both as Hull looked vulnerable early on.
The opener came in the 11th minute, the Tigers scoring with their first attack on the Burnley goal.
Cameron Stewart made a dangerous run down the left flank before firing a low cross into the box for Fryatt, who scored a tap-in for his fifth goal in eight games.
Hull then took control of proceedings, keeping the ball in a more composed fashion than before the goal, but without creating many clear-cut chances.
But just before half-time Fryatt came inches away from doubling the Hull lead, but failed to make contact with Robbie Brady's free-kick.
Sam Vokes came on for Stanislas at half-time as Howe looked to rediscover Burnley's urgency that had faded since the first 10 minutes.
Rodriguez almost obliged for his manager, turning on the spot to strike a ferocious volley at goal, which Peter Gulacsi did well to turn on to the bar.
But as in the first half, it was Hull who took their chance against the run of play.
In the 55th minute Robert Koren found Fryatt in the box, the striker wrapping his foot around the ball to strike high into the net, with Burnley keeper Lee Grant not getting enough on his attempted save.
Fryatt was then released down the left channel and looked set to complete his hat-trick, but his strike was straight at Grant.
Hull looked comfortable but were caught napping with 12 minutes left, as Burnley defender Edgar stole in behind Andy Dawson to nod home a Ross Wallace cross from the right and make it 2-1.
Moments later another ball from the right, this time from Kieran Trippier, again found Edgar, who volleyed high into the net to draw the Clarets level.
There was only going to be one winner in the closing stages as Burnley poured forward looking for the three points, and Rodriguez's composed finish from the right channel gave his side their first win in five matches, and left Barmby tasting defeat for the first time.