It's close to two years since the Clarets drew 0-0 in a League fixture and it might take them that long to figure out how it ended that way at Selhurst Park.
Brian Laws' men produced one of their best performances away from home in recent times but couldn't find a way past inspired goalkeeper Julian Speroni or the Palace woodwork.
Substitute Rodriguez could have had a hat-trick after his early appearance as a substitute gave him the platform to exhibit his growing maturity.
Chris Eagles also found Speroni an impassable object and not surprisingly the Argentinian headed for the dressing room at the end clutching a man-of-the-match bottle of champagne.
Laws had made a significant change to his starting line-up following the mid-week defeat to Middlesbrough with Dean Marney coming in for skipper Graham Alexander in midfield with the immaculate Jack Cork smoothly slotting back into the holding role.
And Rodriguez was soon thrown in the fray when Ross Wallace tweaked a hamstring after just 20 minutes.
Jay Rodriguez lets fly with another effort
Chris Eagles was a constant threat
But far from being disrupted, the Clarets went from strength to strength once they had survived their seemingly customary couple of early scares.
The impressive Eagles set the tone after 25 minutes when his long-range effort was tipped onto the bar by Speroni and Burnley dominated virtually from them on in.
Rodriguez and Speroni then started their personal duel just before the break and it resumed early in the second half when the keeper made a brilliant block to deny the striker after a flowing Clarets' move.
Eagles kept peppering the target and Rodriguez was again thwarted by the Palace number-one with an acrobatic volley from Tyrone Mears' cross.
Strong man Andre Bikey
Inevitably as Burnley kept pressing there were opportunities on the break for the home side and and Jensen earned his corn with an excellent save at the feet of lively substitute Wilfried Zaha before pushing a Neil Danns' shot onto the post in the last minute.
But a Palace winner would have been a travesty and the Clarets felt short-changed by a point after Rodriguez's luck was summed up with a stinging drive which hit the underside of the bar and bounced out 11 minutes from the end of a thrilling game.
Burnley: Jensen, Mears, Fox, Carlisle, Bikey, Marney, Cork, Elliott, Eagles, Iwelumo (Thompson 70), Wallace (Rodriguez 21). Subs: Grant, Alexander, Duff, Paterson, Cort.
Crystal Palace: Speroni, Marrow, Clyne, Davis, McCarthy, Garvan, Dorman (Bennett 62), Cadogan (Zaha ht), Danns, Vaughan, Dijilali (Andrew 83). Subs: Barrett, Price, Bennett, O'Keefe, Counago.
Referee: James Linington.
Booked: Mears 40, Elliott 70, Carlisle 72, Fox 90
Marrow 44, Davies 59
Jack Cork ran the midfield