Victory – secured late on as Ben Toolis went over for the hosts at BT Murrayfield – lifts Solomons’ side back up to fifth in the table – four points behind Ulster.
Phil Burleigh also crossed while Blair Kinghorn kicked four penalties as Edinburgh claimed a narrow victory despite being outscored three tries to two.
And Solomons admitted his side had been pushed to the limit before securing a crucial four points in the battle at the top end of the table.
“That win was absolutely critical for us if we want to compete for a top six spot," he said. "From our squad of 43 we had 18 guys gone, so strength in depth is vital.
"All the players that came on made a difference and it speaks volumes that we can put out a competitive side under these circumstances."
The Welsh region grabbed three first-half tries, through Hadleigh Parkes, James Davies and Steffan Evans, to lead at the break.
However, Edinburgh are not easily beaten at BT Murrayfield – they have lost there just once there in all competitions this season – and a half-time rocket from Solomons did the trick.
"We made it really difficult for ourselves, we gifted them their first 13 points," added Solomons.
"We simply weren't at the races, we were too loose, and we had a few words at half time. I think I broke the rage-ometer!
"But, second half, the players showed enormous resolve. They were not going to lose that game.
"The energy we showed in the last 10 minutes was fantastic, we need to show that for 80 minutes."
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