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But Phil Godman, making his first appearance in the Heineken Cup this season after injury, came off the bench to score an injury time drop-goal at Stade Yves du Manoir to clinch Edinburgh's fourth victory of the campaign.
The win puts them four points clear of the Blues, who play on Saturday but Bradley was already focusing on his side's final pool encounter, against London Irish at Murrayfield next Sunday.
"We do have the advantage as we are playing at home and there is no need to travel," he said.
"I think our fans will be excited about it and the players will and should be excited as well."
"It could be a fantastic day next weekend, but we have to remember when you go across that line its still just a game and we have to do the basics right and we didn't do them all right tonight."
Edinburgh took the lead three times in Paris, with Ross Rennie, Netani Talei and David Denton all touching down, only for Racing to repeatedly level the score.
The difference came down to Godman's late strike and Bradley praised the Scottish fly-half following his return from injury.
"We got lucky at the end, we held our composure and fell over the line with Phil's drop-goal," he added.
"And credit to him, he's put in all the hard yards in training in the last six months to get back and come on a win a game like that.
"Credit has to go to both set of players, it was a good open game of rugby, particularly in that second half.
"Racing really tore into at the start of that period and we had to weather the storm, but we got there in the end."
"It was much different game to that last match against them and it was tighter in a lot of areas.
"But we knew that the side that could hold onto the ball would get an opportunity in the last ten minutes and that's what happened.
"Racing got ahead, then we got back level with the penalty from Laidlaw and then the drop-goal which was perfect timing for us."
