Seneca the Younger frequently contemplated the beginnings and ends of things and the larger meaning of this in multiple letters and essays.
The given sentence is closest in thought to some of the ideas from Naturales Questiones (Natural Questions):
Ergo, quandoque erit terminus rebus humanis, cum partes eius interire debuerint abolerive funditus totae ut de integro totae rudes innoxiaeque generentur ........ peracto exitio generis humani extinctisque pariter feris, in quarum homines ingenia transierant, iterum aquas terra sorbebit, terra pelagus stare aut intra terminos suos furere coget, et reiectus e nostris sedibus in sua secreta pelletur oceanus, et antiquus ordo revocabitur. Omne ex integro animal generabitur1 dabiturque terris homo inscius scelerum et melioribus auspiciis natus.
(Therefore, whenever the end comes for human affairs, when parts of the world must pass away and be abolished utterly so that all may be generated from the beginning again, new and innocent, .... when the destruction of the human race is completed and the wild animals, into whose savagery men will have passed, are equally extinct, again the earth will absorb the waters. The earth will force the sea to stay in its place or to rage within its own boundaries, and the ocean will be ejected from our abode and driven back to its own secret dwelling-place, and the ancient order of things will be re-established. Every living creature will be created anew and the earth will be given men ignorant of sin, and born under better auspices.)
Trans: Thomas H. Corcoran
There is no statement in Seneca's known works where he writes "Every new beginning comes from some other beginning's end." The sentence originated with the song by Semisonic and then was used in various pop inspirational posters that falsely attributed the lyric to Seneca.