Phil Davies’ men went into the Arms Park clash on the back of consecutive defeats to Zebre and Leinster.
But they got back to winning ways thanks to Cuthbert’s brace and 19 points from the ever-reliable boot of Leigh Halfpenny.
The Blues returned to the Arms Park for the first time since their shock defeat at the hands of Zebre and they came flying out of the blocks.
They displayed plenty of determination and Halfpenny opened the scoring on three minutes.
He was uncharacteristically off target minutes later and were fortunate not to concede a try when Edinburgh’s centres combined off the back of a scrum only for Atiga’s pass to fly behind Tim Visser.
Scotland star Greig Laidlaw levelled the scores and then nudged the visitors into the lead with a pair of penalties.
But Halfpenny quickly responded and moments later Cuthbert crossed the whitewash.
Owen Williams created the opening with a delicate grubber kick that Harry Robinson re-gathered before returning the ball to the Wales centre.
Williams was hauled down short but possession was retained and the ball was spread wide for Cuthbert to squeeze in the corner.
Halfpenny converted from the touchline but Laidlaw hit-back with a pair of penalties to cut the Blues’ lead to 13-12 at the break.
Laidlaw struck the uprights within minutes of the restart and Leonardi had a close-range effort ruled out by TMO Gareth Simmonds for a double movement.
But Halfpenny kept the scoreboard ticking and the Blues put Edinburgh to the sword shortly after the hour mark with a sensational second try.
With Edinburgh reduced to 14-men and Joaquin Dominguez hobbling, Cory Allen made the initial break.
Dafydd Hewitt kept the ball alive and Lions duo Halfpenny and Sam Warburton combined to send fellow Australia tourist Cuthbert racing over for his second.
Moments later Rhys Patchell released the red-hot wing to burst clear but he was denied a sensational hat-trick by some equally impressive defensive work from his namesake Jack Cuthbert.
