The Scottish outfit took a big step towards qualification for the knock-out stages, with their 24-6 victory moving them five points clear of second-placed Bordeaux Begles with two games remaining.
They had gone into the break 9-3 ahead thanks to three penalties from Sam Hidalgo-Clyne before two tries from Dougie Fife and Roddy Grant further broke down the stubborn London Welsh side at the Kassam Stadium.
Attention now turns back to the GUINNESS PRO12 with Edinburgh playing host to Benetton Treviso on Friday night.
Compared to Europe, it has been a mixed bag for Solomons' men in the league this season with three wins and five defeats.
But with success breeding confidence, Solomons is optimistic they can transfer their form into the league.
"What these wins in the Challenge Cup have proved is that our strength in depth is much, much better than it has been in the past," Solomons told the Edinburgh News.
"We are playing without a number of international and front-line players, yet we are still managing to put in some strong performances.
"The first thing is always to win the game. That's important and that's what we did.
"What was also very positive was the way in which our tight five dominated the scrum, line-out and maul. They were very good and meant we were buried in their territory in the second half.
"If we can keep on developing a winning culture and mentality then the confidence in the squad will grow.
"If we win our next two games, then we'll be through to the quarter-finals. That would be a significant step forward for this group to reach the last eight in a European tournament."
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