Hamish Watson scored their only try, early in the second half, while Sam Hidalgo-Clyne and Blair Kinghorn provided the rest of the points from the kicking tee.
And after a tricky spell during the Six Nations, head coach Solomons is aiming for a strong finish to the season as Edinburgh battle for a place in the top six.
Solomons said: "We really made it difficult for ourselves. There were some defensive lapses, we made critical errors but the important thing was that we hung in there and got the win.
"Nobody who comes here gets an easy game, it's not an easy ground to come to.
"We've struggled over the Six Nations, without a doubt. We had five of our leaders, our big guys out but that's also enabled our youngsters to get experience they perhaps wouldn't have got which will stand them in good stead for next season.
"We're such a tight group - everyone wants us to do well, wants the club to do well."
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