Sean Dyche's side thought they had secured all three points when substitute Junior Stanislas rifled home from 25-yards in the 76th minute.
But a defensive error with five minutes remaining silenced Turf Moor. A poor backpass from Scott Arfield led to Michael Duff clearing against Rhodes as the ball looped over Tom Heaton.
The result extends Burnley's 34-year run without a victory against their rivals.
Elsewhere, Blackpool's fantastic start to the season continued as Paul Ince's side triumphed 2-1 away to Bournemouth at Seward Stadium.
Ricardo Fuller opened the scoring in the seventh minute, but Lewis Grabban levelled just five minutes later.
But Blackpool were not to be denied the points and Neal Bishop notched shortly after the restart to maintain their unbeaten start to the season.
QPR have also yet to taste defeat in the Championship and added another win to their tally, with a 1-0 success at home to Birmingham City at Loftus Road.
Charlie Austin grabbed his first league goal for the club in the 49th minute to ensure that Harry Redknapp's side gained all three points.
Derby County embarrassed Millwall 5-1 at The Den, as Nigel Clough's side secured their third victory on the road in as many away games this season.
Jake Buxton, and two from Craig Bryson put Derby 3-0 ahead before Martyn Waghorn pulled one back.
Bryson grabbed his hat-trick in the 81st minute, before Mason Bennett completed the rout.
The match was stopped temporarily due to apparent crowd trouble in the home end, amid reports that one supporter had an altercation with Clough.
Elsewhere, Sheffield Wednesday and Bolton Wanderers are still looking for their first league wins this term. The former were held 1-1 at Hillsborough by Yeovil Town, while the latter succumbed to a 1-0 loss to Leeds United at the Reebok .
Leicester City moved up to fourth place after beating Wigan Athletic 2-0 at the King Power Stadium.
Liam Moore notched the opener in the 15th minute, before a David Nugent penalty in the second half gave Nigel Pearson's side a deserved victory.
Nottingham Forest were pegged back twice by Barnsley at the City Ground, but Darius Henderson's late intervention secured a 3-2 triumph.
A Chris O'Grady penalty cancelled out Simon Cox's opener before Kelvin Etuhu's own goal put Forest ahead for a second time and then Tomasz Cywka once more levelled proceedings.
Henderson was the hero in the 83rd minute for Billy Davies' side.
In other matches, Ipswich Town defeated Middlesbrough 3-1, Huddersfield Town and Doncaster Rovers played out a 0-0 stalemate and Watford were held to a 1-1 draw by Charlton Athletic.