But this season Edinburgh were able to battle back from a half-time deficit thanks to the boot of Greig Laidlaw and then try machine Tim Visser sped 40 metres up the left touchline to grab a late matchwinning score.
"We know we have got a long way to go, we are under no illusions on that. But to me, it is important that we keep improving week by week," said Solomons.
"It was great for the guys that they got that result. It was a real boost to their confidence. Of course that will help them take their next step up. But what we have been saying is that we need to be performance focused - not result-focused.
"If we get our performances right, and we have been so close, we will be okay. When we go through the match review on a Monday, we have been saying to the guys you are so close you won't believe it.
"I'm a massive believer that the difference between success and failure is the finest line in the world. If you persevere, you will cross that line. We are a determined lot and we are most certainly going to persevere."
Laidlaw contributed a magnificent 19-point haul in the Murrayfield victory. And while Solomons was reluctant to single out individuals, he could not resist laying the praise on his in-form scrum-half.
"I don't want to take anything away from anyone in the team because I felt everybody played well. But I thought Greig led the side really, really well. He has a very good temperament and showed that with his kicking to the posts," added Solomons.
Edinburgh now face a daunting trip to Stade Aimé Giral to face Perpignan, who were edged out by Gloucester in another Pool 6 thriller, in Round Two.