The Swiss took the lead after 28 minutes when Nadine Riesen rifled an effort in off the near post - an advantage they deserved after a promising first half during which Gerladine Reuteler smashed the crossbar amid a raucous atmosphere in Basel.
It was all change after the break, however, with Hegerberg drawing Norway level on 54 minutes when she beat Chelsea goalkeeper Livia Peng to a corner and powered home a header. Just four minutes later, Gemma Grainger's side took the lead as Switzerland defender Julia Stierli turned into her own net.
Hegerberg missed the opportunity to seal the victory when dragging a penalty wide of the post after a handball in the box, and moments later Switzerland had a spot-kick award of their own overturned by an automated offside decision.
Cecilie Fiskerstrand was called into action on a couple of occasions in the Norway goal but they held on and go top of Group A.