The defender, who thought he had earned the home side a point before Ross Caldwell earned the visitors the 2-1 victory, believes it will take "years" of success before Hibs can claim to be kings of the capital.
This win was just Hibs' first Clydesdale Bank Premier League victory at the home of Hearts in four years but the second Hibs have enjoyed over their old foes this term after they knocked them out of the William Hill Scottish Cup in January.
This season's three other SPL meetings ended in draws, leaving Hearts unable to muster a league victory in the derby for the first time since the 2000/01 season.
And Caldwell says it is those facts which show the Leith side can now rightly claim to be top dogs in the city.
He said: "No I don't think the balance is shifting. That's just talk. If you watched that game you will have seen there was nothing in it.
"How many chances did they create really? It would take them years of finishing above us before they can say they are the dominant side."
Barr, meanwhile, will soon be on the move after being told his contract at Hearts is not being renewed. Losing his final game at Tynecastle, therefore, was not how he wanted to sign off after three years in maroon.
The defender - whose goal was his first since his strike in last May's 5-1 cup final rout of Hibs - said: "It's always sore when you lose a game like that right at the end. The fact that's the first time we lost to them in the league in four years makes it even worse.
"It's my last game at Tynecastle and it was horrible. It's never nice to go out to a defeat to your rivals.
"I was hoping I could go out on a high after getting the goal but it wasn't to be. I've loved my time here but that was a sore one."
- Sports & Recreation
- Ross Caldwell