Question: If a person forgets and says the salam by mistake before completing his salat, will his salat not become invalid?
ANSWER
If a person forgets and says the salam by mistake before completing his salat, he should stand up and continue his salat if he remembers it after saying the salam. He should perform sajda as-sahw (prostration of forgetfulness) at the end of the salat. It is written as follows on the 352nd page of the translation of Halabi as-saghir:

“If a person says the salam at the end of the second rak’at thinking that he has completed Salat az-Zuhr and shortly afterwards remembers that he performed only two rak’ats, he should complete his salat and then do sajda as-sahw, for he said the salam by mistake.”

It is written in the book Ni’mat-i İslam, which is the translation of hashiyah of Maraqi al-falah, in its section dealing with sajda as-sahw:
“If a person, while performing a four-rak’at or three-rak’at fard salat, says the salam thinking that he has completed the salat, he, if he has not done anything that breaks salat, should stand up immediately, complete the salat, and then do sajda as-sahw because of his mistake.”

It is not written in any fiqh book that saying the salam by mistake breaks salat. Salat is not invalidated unless one of the nullifiers of salat, e.g., talking, eating or drinking something, occurs.

Question: While I was offering a four-rak’at fard salat, I mistook the first sitting for the second sitting and said the salam. I remembered shortly afterwards that I had made a mistake, so I stoop up right away and continued my salat. Then I did sajda as-sahw at the end of the salat. Is my salat valid?
ANSWER
Yes, it is valid.

If such a person remembers it after reciting the prayer “Allahumma antas salam …,” his salat becomes invalid.