Solomons, looking to end his first season in charge on a positive note, saw his side lead 6-5 at the break thanks to two Greig Laidlaw penalties.
Tries from Jordi Murphy and Dave Kearney then edged the hosts in front but Tim Visser's late try sent nerves jangling at the RDS.
It was the winger's sixth try of the season and his 60th for the club in 104 appearances but Edinburgh were unable to force the victory in the final 15 minutes, eventually losing out 15-13.
It leaves Edinburgh finishing eighth in the table while table-toppers Leinster host Ulster in next week's semi-finals.
But despite the loss, Solomons insisted there were plenty of positives to take away from their season-ending performance.
"I am extremely proud of the boys," Solomons told the Scotsman. "They are fatigued, we have had a lot of injuries and to come here and have Leinster winding down the clock at the end is a credit to their efforts."
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