But the 50-year-old Irishman insisted his side had the chances to end their losing streak, with replacement wing Tom Brown guilty of a late knock-on with the try line gaping.
"I think that we forced the play, we didn't need to force the play we just needed to stay up a fraction longer in contact, support the ball carrier and I think we would have won that game," said Bradley.
"I think in the first half we had a four against three and the ball got passed in to touch, missed a couple of kicks, perhaps should have been awarded a penalty for the scrum, two turnover lineouts in the last two plays of the match and we lost by one point and that's taking out the overplaying of the ball when we did have it.
"It's not acceptable from our fans' point of view that we lost this match today, and we should be very cross with ourselves, but that's where we are and we'll build on that."
Man of the match Rhys Patchell opened the scoring for Cardiff with an early penalty, making it 6-0 with the boot on 13 minutes. Edinburgh replied when Gregor Hunter succeeded with his third kick at goal, but fly-half Patchell replied in kind to make it 9-3.
And despite a second Hunter penalty, the Blues extended their lead before the break when Patchell beat his man and fed Mike Paterson to touch down.
Trailing 14-6 at the interval prop WP Nel burrowed over from short range on 50 minutes to bring Edinburgh back into contention with a try converted by Hunter.
But despite a Harry Leonard penalty, Patchell kept his cool to dispatch his fourth penalty of the evening and send Cardiff home with the points.
After the game, Bradley expressed his annoyance that what he perceived as Edinburgh's dominance at the scrum wasn't rewarded by the referee, but poor results aside still believes his side have the building blocks in place to improve next season.
"I think all year the sides that have been under that sort of pressure like Cardiff were today have been penalised and it happened twice and it was obviously a disappointment for us," he said.
"In terms of the squad that's there next year, I think there's one or two key signings we need and when they are in place I think Edinburgh will be in pretty good shape."
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