The Manchester City striker hit two goals in seven second-half minutes to take his tally in qualifying to six.
Alexander Sorloth and Fredrik Aursnes were also on target in Larnaca.
Norway face Spain in Oslo on Sunday and a failure to win would ensure Scotland's qualification.
Stale Solbakken's side are third in Group A, five points behind leaders Scotland and two points behind Spain, who have a game in hand.
Scotland could have secured qualification themselves with a point against Spain on Thursday but they were beaten 2-0 in Seville.
Cyprus have lost all six of their Euro 2024 qualifiers.
