TL;DR Not really. In Soviet times there was a joke about the tendency of Soviet scholars to trace the origins of everything to Russian contributions: Russia is the motherland of elephants. I think this is a case of that with Russia replaced by Persia.
Alakbarli is himself an academic, a doctor of historical sciences (highest Soviet degree), who studied Tusi's manuscipts since 1986. The OP linked webpage is a redux of his publication in Russian that gives a fuller quote and the citation to a source:
Xace Nəsirəddin Tusi. Əxlaqi-Nasiri. Baki, Elm, 1989, pp. 47-48. Other more recent works that I have seen either reproduce Alakbarli's (badly translated) Tusi quote or simply repeat the claim phrased similarly to his. Here is the quote Google translated from Russian (with my minor edits):
"If someone, through the eyes of an attentive and inquisitive researcher, pays attention to the process of changing substances, to their composition, transformations and opposites, ... then he will understand that not a single body is completely destroyed. It only changes its form, state, composition, color [and other] qualities and turns into a compound or elementary substance... Because material bodies do not have the property of being destructible".
Further quotes outline Tusi's natural philosophy with changeable "elements with properties of prime matter" at the basis.
It is plausible that the quote is authentic, but, pace Alakbarli's enthusiasm for the fellow countryman (taking that broadly), it is not as remarkable as he makes it out to be. The prime matter and the elements are Aristotelian conceptions, ex nihilo nihil fit was a slogan of ancient Greek natural philosophy, as recorded in Aristotle's physics where it is credited to Parmenides. Even clearer application of it to matter specifically is found in atomists since Democritus, and we can read similar passages in De Rerum Natura. So if this vague philosophizing (that does not even mention mass) counts as "stating the principle" then Tusi was beaten to it by over a millennium and a half.
This said, Whitaker in An Historical Note
on the Conservation of Mass shows that Lomonosov and Lavoisier had precursors even when we take the conservation law in a more quantitative sense relevant to chemistry. Jean Rey in “On an Inquiry Into the Cause
Wherefore Tin and Lead Increase in Weight on Calcination” (1630) stated:
"weight is so closely united to the primary matter of the elements that they can never be deprived of it. The weight with which each portion of matter was endued at the cradle, will be carried by it to the grave. In whatever place, in whatever form, to whatever volume it may be reduced, the same weight always persists."
There are still Aristotelian prime matter and elements in it, but there is also weight and references to actual measurements. Rey then dutifully reports that his measurements did not agree with his philosophizing:"I... affirm that the examination of weights which is made by the balance differs greatly from that which is made by the reason". The contribution of Lomonosov and Lavoisier should now be clear, they converted a vague armchair speculation into a confirmed quantitative law. Democritus, Aristotle and Tusi did not even get to the problem Rey discovered. Calcination went on to become a paradigm case for Stahl's phlogiston theory that Lavoisier displaced.