The Scottish side had the chance to claim victory with an attacking lineout right at the death but the Blues, playing their first game since the departure of Mark Hammett, held firm for their first win in four.
Solomons' side are in a three-way battle with Connacht and Scarlets to claim a top-six finish but defeat at the BT Sport Cardiff Arms Park comes hot on the heels of a narrow defeat at home to Ulster the week before.
Two losing bonus points from two games they could have won angers Solomons - whose side did cross for two tries through Tim Visser and Roddy Grant.
"For the second week in a row we have only ourselves to blame for losing that game," he said.
"We have come away with a losing bonus point, but it should have been four points for the win.
"We missed a kick from in front of the posts and should never have conceded a try just before half-time. All their tries came from system errors in our defence and that is something we have got to sharpen up on.
"It is very, very disappointing and that has made it very, very tough for us to finish in the top six.
"We have got to go away to Treviso next weekend and produce a performance."
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