Chasing a first win over their fierce rivals since 1979, substitute Junior Stanislas looked set to write himself into Clarets folklore with a wonderful 25-yard goal 16 minutes from time.
However, following a wayward pass from Scott Arfield, Rhodes ensured the spoils were shared after Michael Duff's clearance ricocheted off the Scotland international's shin and went in five minutes from time.
Blackburn's Lee Williamson was then dismissed in stoppage time for hauling down Danny Ings on a breakaway attack.
The pressure of the occasion almost got to Rovers goalkeeper Jake Kean in the early exchanges. First he flapped at a Kieran Trippier cross before breathing a huge sigh of relief when his poor clearance was charged down by Sam Vokes only for the ball to roll wide.
After a Trippier free-kick flew narrowly over, Kean was again grateful to see Tommy Spurr divert the flying full-back's low cross inches above his bar.
And Kean was given no respite as Arfield and the in-form Ings drew him into successive near-post stops.
Indeed it took 22 minutes for the visitors to muster their first effort, Rhodes, wearing a cast on his broken hand, steering well off target.
Gary Bowyer's men posed a threat on the break and Clarets stopper Tom Heaton did brilliantly to push Tom Cairney's left-foot try past the post.
In the second half, Bowyer's backroom staff were left incensed when referee Craig Pawson stopped play to book Clarets centre-back Duff, yet Rovers wanted an advantage as Rhodes was in the clear.
They almost drew first blood anyway moments later courtesy of centre-back Grant Hanley.
From the fourth of five consecutive corners, Hanley's first header was pushed on to the top of the bar by Heaton and, from the resulting centre, the Scotland international somehow failed to get a good connection from six yards.
Burnley would, though, seize the initiative when boss Sean Dyche introduced Stanislas in the 68th minute.
It took just eight minutes for him to make an impact as he played a wonderful one-two with Ings before unleashing an unstoppable drive home.
Yet just as the hosts were beginning to contemplate a historic victory, Rhodes found a sniff of goal.
The striker pounced on Arfield's lapse and turned inside Jason Shackell where Duff was waiting to avert the danger, however he hammered the ball against Rhodes' leg and the ball looped over Heaton and in.
It provided the second straight heartbreaking 1-1 draw for Dyche's men against Rovers following David Dunn's offside goal at Ewood Park in stoppage time back in March.
In this game's added time, last-man Williamson took a savvy dismissal which may well have saved his side a point for hauling down Ings on the halfway line.