Law, who started just off John Sutton in his first appearance since a summer move from Conference side Alfreton Town, showed early promise in his bid to emulate brother Nicky's impressive two-year spell at Fir Park as he finished twice after getting ahead of the visiting defence.
Motherwell were well on top until the 35th minute when Olafur Karl Finsen converted a penalty after Keith Lasley was adjudged to have handled.
And they suffered a huge blow in injury-time when Finsen netted again after Lasley was ruled to have pushed over Rolf Toft as Motherwell scrambled to defend a counter-attack.
The hosts might have had a penalty themselves seconds earlier when Simon Ramsden went down as he looked set to score following a corner.
Stjarnan, who sit second in the Icelandic League after 11 unbeaten games, have the advantage of playing on their own artificial pitch for the second leg of the Europa League second qualifying round.