There are few things at play here, Carol absorbs the energy from the light speed engine that powered the ship not from the Tesseract itself. This energy is then likely less fatal than if she'd absorbed power directly from the Space Stone.
Talos: She wanted you to help us find the core.
Carol: Well, I already destroyed it.
Talos: No, you destroyed the engine. The core that powered it is in a remote location.
Even if she did get the energy directly from the Tesseract, it was specifically designed so that "inferior species" could use the Infinity Stone inside of it. Therefore, the energy from the Space Stone was already safer than if it came directly from it.
...when an inferior species makes direct contact with the Stones, they’re ripped apart by the sheer force of the thing. It’s a glorious and frightening sight. In order to protect these beings, the Stones were given defensive casings: the Orb for the Power Stone, the Scepter for the Mind Stone, the Tesseract for the Space Stone, the Eye of Agamotto for the Time Stone, and”—his voice hardened—“as you know, the Aether for the Reality Stone.”
Infinity War: The Cosmic Quest Vol.1
Lastly, after the accident Yon-Rogg gives his Kree blood to her and Kree blood has regenerative capabilities. This is assuming he’s telling the truth which he appears to be as Carol did have blue blood in the training scene.
Yon-Rogg: That's my blood that's coursing through her veins.
Agent Coulson was injected with that, and literally minutes later, his wounds showed signs of cellular regeneration.
Agents of S.H.I.E.L.D., Season 2 Episode 7, "The Writing on the Wall"
It's also worth noting that the Kree appear to be able to hold the Infinity Stones perfectly fine without being ripped apart, see Ronan in Guardians of the Galaxy, so having their blood would certainly have helped her after the fact.