The Argentina forward's creativity has the Turkish champions on the brink of a return to the lucrative group stage for the first time in four years.
Icardi's standout moment on Wednesday night was a deft volley to score in the 29th minute, taking a floated pass from Yunus Akgun over his right shoulder with a first-time shot to give the visitors a 2-1 lead.
Molde had led in the fifth minute and levelled early in the second half, but were beaten in the final seconds. Icardi broke clear and passed across the goalmouth for Norwegian midfielder Fredrik Midtsjo to score into an empty net.
Midtsjo's goal threatens to extend Norway's unwanted record of not having a team advance to the group stage since his former club Rosenborg 16 years ago.
The return game is in Istanbul next Tuesday between two teams that already each won through two qualifying rounds.
Braga will take a 2-1 lead to Greece after Panathinaikos found a goal in the 95th minute to halve the Portuguese team's lead.
Midfielder Daniel Mancini's goal gave hope to Panathinaikos, the 1971 European Cup beaten finalists that have not been in the group stage for 13 years.
Braga got goals in the 51st by Abel Ruiz and in the 73rd from Spanish winger Alvaro Djalo, either side of a glaring miss by Panathinaikos forward Andraz Sporar who scooped a shot over the unguarded goal.
And in Israel, Maccabi Haifa were held to a 0-0 draw by Young Boys in a game of few chances where the visiting Swiss champions did not have a shot on target.
Winners in the second legs advance to the 32-team group-stage draw on August 31.
