Does a person have a glimpse of Nibbana in a moment of true mindfulness and the three fetters drop away?
Yes, this is the correct explanation. The path of purification is as follows:
- The meditator undertakes the practice of morality, specifically guarding the mind from unwholesomeness (sīlavisuddhi)
- Guarding the mind, the meditator cultivates focus, gaining clarity of mind based on the objects of experience (cittavisuddhi)
- Having a clear mind, the meditator cultivates an understanding of the nature of experience as composed of impersonal physical and mental constituents (diṭṭhivisuddhi)
- Observing the physical and mental phenomena, the meditator cultivates an understanding of the causal interactions between the physical and mental phenomena (kaṅkhāvitaraṇavisuddhi)
- Through the understanding of positive and negative causal relationships, the meditator cultivates an understanding of what is an what is not the path (maggāmaggañāṅadassanavisuddhi)
- Through an understanding of the path, the meditator cultivates the right path (paṭipadāñāṅadassanavisuddhi)
- Through cultivating the right path, one attains knowledge and vision of the noble path and fruition (ñāṅadassanavisuddhi)
Sotāpanna occurs upon attainment of the seventh stage. The right path (#6) is the gradual understanding that all formations are impermanent, suffering, and non-self. Once this realization becomes all-encompassing, the meditator attains an absolute certainty of one or another of the three characteristics and this leads to a release based either on knowledge of signlessness (based on impermanence - that there is no telling what will happen in advance), desirelessness (based on suffering - that there is no benefit to clinging to any formation), or emptiness (based on non-self - that all formations are void of self and there is no relationship of ownership or control in regards to all formations).
This release leads to an experience of cessation, where there is no arising of sense experience (including mental sense experience). This is the realization of nibbāna, and this is what leads to the eradication of the first three fetters.
Wrong view is eradicated because one can never believe that anything could be permanent, satisfying or controllable, having seen them all cease without remainder.
Attachment to wrong practice is eradicated because one can never be confused about the practice that leads to nibbāna after seeing nibbāna for oneself.
Doubt about the Buddha, Dhamma or Sangha can never arise, because one knows what the Buddha taught to be true with complete certainty, and the results that one attains having followed said teachings.
Here are some source quotes from the Visuddhimagga:
Then, while every sign and occurrence appears to him as an
impediment,when conformity knowledge’s repetition has ended,
change-of-lineage knowledge arises in him, which takes as its object
the signless, non- occurrence, non-formation, cessation,
Nibbāna,—which knowledge passes out of the lineage, the category, the
plane, of the ordinary man and enters the lineage, the category, the
plane, of the Noble Ones,—which, being the first adverting, the first
concern, the first reaction, to Nibbāna as object, fulfils the state
of a condition for the path in six ways, as proximity, 
contiguity, repetition, decisive-support, absence, and disappearance
conditions,—which is the culminating peak of insight,—which is
The abandoning of the states beginning with the fetters by the
noblepath knowledge in such a way that they never occur again, like a
tree struck by a thunderbolt, is called abandoning by cutting off.
With reference to this it is said: “Abandoning by cutting off comes
about in one who develops the supramundane path that leads to the
destruction [of defilements]” (Paṭis I 27).
(Both from Nyanamoli's translation of Vism XXII)