The Sky Blues sacked boss Aidy Boothroyd on Monday following a run of one win in 16 games, but they dominated for long periods under the temporary stewardship of chief scout Andy Thorn.
Rodriguez displayed excellent control and an assured finish to open the scoring after 18 minutes, but Coventry deservedly levelled through Lukas Jutkiewicz shortly after the hour.
A splendid solo effort from Gary McSheffrey looked to have given City the win, but Rodriguez headed his 12th of the season 11 minutes from time.
The result leaves Burnley three points shy of sixth-placed Nottingham Forest with a game in hand, while Coventry edge eight points clear of the bottom three.
Eddie Howe kept faith with the Burnley XI that lost 3-0 to Millwall at the weekend, while Coventry made three changes to the side that went down 1-0 to Hull in Boothroyd's final game at the helm.
Stephen O'Halloran and Jordan Clarke came into the Sky Blues' defence for Nathan Cameron and the injured Martin Cranie, while Carl Baker replaced David Bell in midfield.
Coventry started brightly and in the third minute Jutkiewicz hit a sumptuous volley on the turn that drew a magnificent one-handed save from Burnley goalkeeper Lee Grant, before fellow forward Freddy Eastwood skipped past full-back Danny Fox and drilled into the side netting.
Burnley were disjointed in the opening stages but took the lead through a moment of quality.
Tyrone Mears found Rodriguez 30 yards from goal, where a clever touch took the England Under-21 forward clear of centre-back Richard Keogh and gave him space to coolly lob onrushing keeper Keiren Westwood.
Westwood then held efforts from Nathan Delfouneso and Rodriguez, but Burnley attacks were becoming increasingly sporadic and they lost midfielder Dean Marney to injury five minutes before the break.
His replacement Wade Elliot was easily passed as Baker cut in from the left and unleashed a rasping 25-yard strike that brushed the side netting with Grant beaten.
City started strongly again after the break and Grant got down well to palm away Sammy Clingan's 52nd-minute free-kick.
Coventry should have been level when Eastwood danced past a pair of challenges and touched the ball towards Jutkiewicz in the area.
The former Swindon man momentarily lost his footing, allowing Grant to spring from his line and thwart the eventual attempt.
But Jutkiewicz made amends by bringing Coventry level in the 61st minute, heading home at the second attempt after his initial attempt from Baker's cross had thudded into the crossbar.
Coventry continued to press and skipper Richard Wood glanced a corner narrowly wide, before the lively Baker jinked past the Burnley defence from the left and slotted his shot a fraction past the near post.
A deserved lead arrived for the visitors in spectacular fashion with 15 minutes remaining. McSheffrey beat Jack Cork to a loose ball on the right flank, turned into the box to evade two more challenges and curled a delicate left-footed effort beyond Grant into the top corner.
However Burnley hit back in the 79th minute when Rodriguez headed home from a Fox corner despite the best efforts of Aron Gunnarson to clear.