The USA picked up its third straight shutout in a contest that saw both teams have good scoring opportunities.
The lone goal came in the 21st minute when Amy Rodriguez, with her back to the goal, spun a first-time pass to Rapinoe, who beat a defender and smashed a skimming shot across the face of the goal, off the left post and into the net.
Goalkeeper Hope Solo was called upon to make several big saves, most notably in the 18th minute with a dive to tip Kristine Lie's shot over the goal with her right hand and again in the 83rd minute when she stood straight up and tipped Norwegian captain Ingvild Stensland's rocket off the crossbar and over the goal.
The game marked the international debut for U.S. defender Kendall Fletcher.
The USA will be looking for its sixth Algarve Cup title in the last seven years. It fell in penalty kicks to Germany in the 2006 final.
March 9 in Ferreiras, Portugal
USA 1 Norway 0. Goal: Rapinoe 21.
USA -- Solo, Mitts (Boxx, 62), Rampone, Fletcher, Buehler, O'Reilly (Woznuk, 83), Hucles (Lloyd, 46), Tarpley, White, Rapinoe (DiMartino, 70), Rodriguez (Kai, 46).
Norway -- Knutsen, Huse (Wiik, 60), Ronning, Gardsjord, Akerhaugen, Stensland, Giske, Mjelde, Storlokken (Gronli, 80), Isaksen (Kaurin, 60), Lie.
Yellow card: USA -- Fletcher 45; Norway -- Ronning 75.
Referee: Gyongyi Gall (Hungary).