And after just one win from their previous five RaboDirect PRO12 encounters, Edinburgh duly seized the chance to haul themselves off the bottom of the table at Murrayfield.
Solomons, whose side beat Munster and put up a decent display in defeat to Perpignan in Europe, welcomed back several key players from injury and they combined to make an almost instant impact.
Full-back Greig Tonks, in his first start in a month after hamstring problems, opened the scoring in the 11th minute following a flowing backs move that was initiated when Georgian international Dimitri Basilaia, back from a groin injury, broke the game down.
Benetton Treviso had won their last two matches against Edinburgh - including a 27-22 win at Murrayfield last December - and were always in the match, the influential Nick De Luca's late tackle saving what appeared an almost certain try in the 15th minute.
Edinburgh scrum-half Greig Laidlaw exchanged missed and successful penalties with James Ambrosini as the first half ended 10-3 and the game certainly improved after the break.
Laidlaw edged the hosts further ahead with another penalty following the restart before Treviso finally started to fire.
And after a period of sustained pressure and possession, replacement and Italian international Luke McLean scored in the 61st minute and Tobias Botes converted before then levelling the scores with a penalty.
But roared on by a crowd of 3,993, the hosts' fighting spirit finally got them over the line and eased the pressure on Solomons, with lowly Zebre the next visitors to Murrayfield.
In fairness Edinburgh were playing good rugby, while lacking the final touch, and were due some luck this season.
And when De Luca and Jack Cuthbert chased after Laidlaw's well-aimed kick, bundling Treviso's last man into touch on the five metre line, the foundation was established.
However, it was prop WP Nel who bulldozed over from short range after showing good composure at the line out, Laidlaw then easing the nerves with a well-threaded conversion.
But Treviso still surged forward in pursuit of the levelling score, stretching Edinburgh's determined defence across the pitch but just failing to breach their resolve.
|Edinburgh Rugby Score Card|
|Benetton Treviso Score Card|