The Scots went into the Plate competition after losing their Cup quarter-final to eventual champions New Zealand.
Scotland won their first ever victory against Fiji - in the ninth minute of sudden death extra time - to reach the final of the Plate.
With the Scots Leading 14-7 in the dying seconds of their Plate semi final, Fiji captain Marika Vunibaka found space in midfield and sprinted clear of the despairing Scotland defence to level at 14-14.
In sudden death extra time both sides had good chances to clinch victory, Jim Thompson making a superb tackle on the outstanding Setefano Cakau and Iliavi Satala denying Mike Adamson before Scotland worked space out wide to put in Ben Addison for the decisive score and send the Murrayfield crowd into mass hysteria.
Scotland coach Stephen Gemmell was proud of their display.
"I'm very disappointed not to come out with silverware, our performance in the final maybe merited it," he said.
"The effort has been magnificent, we have just equalled the best a Scotland sevens team has done."