John Hardie's late try proved decisive for Edinburgh, who lost second row Anton Bresler in the first half and found themselves stretched because of injuries.
And having gone down in the reverse fixture last week, Solomons was delighted to get the four points for the win.
"The most important thing was simply to win the match, especially after last week's result at London Irish," he said.
"What was most pleasing was the way the team came back from difficult circumstances to get the win. We were already without a number of key players in the forwards when we lost Anton Bresler just before half-time. We also had injuries to Matt Scott and Andries Strauss.
"We spent a lot of the game in our own territory and didn't put Irish back in their own half. We also made a number of errors when we had opportunities but overall we are just happy with the win."
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