Fetu'u Vainikolo crossed for Eric Elwood's men in the first half who led 10-0 at the interval, Niall O'Connor adding five points with the boot.
But the Gunners improved after the break as the RaboDirect PRO12's top try scorer Tim Visser and Chris Paterson both crossed but three penalties for the hosts along with a last-gasp Mike McCarthy score secured their first league win since September.
Edinburgh's slump continues however with five successive defeats in the league, despite an impressive berth in the quarter-finals of the Heineken Cup.
And Bradley admitted that his troops will need to put together a performance for the entire 80 minutes if they want their domestic form to improve.
"We had a very up and down performance," said Bradley.
"When we were good we were very good, and picked up a good score even when a man down but, equally, when we were poor we made life easy for Connacht.
"Credit to them, over the piece they were the most consistent side, particularly at the breakdown, and that was one of the key reasons for us not getting a win."
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