Murphy struck first for the young Scots but they could not hold on
Scotland will have to overcome a narrow first leg deficit at Easter Road on Monday if they are to reach the finals of the Under-21 European Championship.
Motherwell striker Jamie Murphy fired Billy Stark's visitors ahead on 19 minutes in Iceland.
However, a home side that included seven full internationals hit back through Johann Berg Gudmundsson before the half time interval.
And Almar Ormarsson netted the winning goal on 78 minutes.
The Scots will be disappointed to have thrown away a lead but know a 1-0 win in the second leg will see them into the finals for the first time since 1996.
The evening did not start well for Iceland, who had Rurik Gislason ruled out through injury and then saw goalkeeper Haraldur Bjornsson suffer an injury in the warm up.
That meant 19-year-old Arnar Darri Petursson was handed a place in the starting XI, but Scotland could only manage two shots on target in the whole game.
After a cagey opening, Scotland goalkeeper Alan Martin had to save twice in the 15th minute.
Gylfi Sigurdsson and Birkir Bjarnason opened up the visiting defence and Kolbeinn Sigtorsson fired in a shot from the near post which Martin saved before also blocking the follow-up from Sigurdsson on the penalty spot.
Scotland took the lead on a rare break when defender Holmar Eyjolfsson lost the ball in his own penalty area and Murphy hammered the ball into the net from the right-hand side of the penalty box.
Iceland's response came 15 minutes later. Bjarnason's right-wing cross was cleared only as far as Gudmundsson, who equalised with a great shot with his left foot from outside the area.
The goalscorer was dangerous on the left wing in the second half, producing several good crosses that Kolbeinn Sigtorsson failed to take advantage of, twice heading over the bar in the opening 15 minutes.
But Ormarsson was far more accurate in the 78th minute after coming on as a substitute just five minutes earlier, firing in a superb winner from 30 yards to give his side hope of reaching the finals for the first time.
Iceland U21: Petursson, Valgardsson, Eyjolfsson, Gunnarsson, Eggert Jonsson, Bjarnason (Ormarsson 73), Fridgeirsson, Vidarsson, Sigthorsson, Sigurdsson, Gudmundsson. Subs Not Used: Ingason, Sigurdarson, Kristjansson, Finnbogason, Kristinn Jonsson, Palsson.
Booked: Eyjolfsson, Fridgeirsson. Goals: Gudmundsson 34, Ormarsson 78.
Scotland U21: Martin, Caddis, Scobbie, Wilson, Hanlon, Arfield, Wotherspoon (Goodwillie 55), McGinn, Maguire, Murphy (Templeton 85), Bannan. Subs Not Used: Gallacher, Perry, Marr, Griffiths, Shinnie.
Booked: Arfield, Goodwillie. Goal: Murphy 19.