The only trouble with this is that it is wholly untrue. There is no linguist in the world who would identify Old English as the same language as English, if she met them without knowing their historical relationship. Old English is no more English than Latin is French. The two languages are not mutually intelligible. You could make a much stronger case for Dutch being the same language as English than you can for Old English: its grammar and vocabulary are much closer.
Well, giving up that position, you could retreat to this: it may not be the same language, exactly, but it's the same literature. Middle English literature grew out of Old English literature: there's an organic relationship between the two.
But in fact there is no literary relationship. The Middle English poets, so far as we know, were utterly ignorant of Old English. There is no evidence in the whole corpus of Middle English literature of a single reference to Beowulf or any other Old English poem. Old English literature did not grow into Middle English literature. It was completely destroyed by the Norman conquest. It was as dead, by the year 1300, as Hittite or Akkadian. The first important English poet who could read any Old English – and he certainly didn't read Beowulf – was Milton, in the 17th Century. You could speculate that there was a living ballad tradition, which both the sophisticated Old English poets, with their utterly different metrics, and the Middle English poets, with their French verse-forms, were familiar, although the Old English poets never mentioned it and Chaucer mocks the ballads with devastating parodies – but we could speculate all kinds of things. Perhaps they all studied message-in-a-bottle Chinese poetry too, without ever mentioning it. This is not the stuff of literary history.
There is a historical linguistic relationship. The West Saxon dialect of Beowulf is a sort of linguistic uncle (not father) to the Middle English of Chaucer's London. Old English did grow into Middle English, and if you believe that a literature mystically transmits its national soul, by way of the blood of language – and people after all have believed much sillier things -- then you can say that Beowulf begot the Canterbury Tales. But that's the only demonstrable relationship there is. The only thing they have in common beyond that is that they were written on the same island.
The truth of the matter is that the tradition of English poetry begins in the 14th Century, with Chaucer and Gower. And their mentors were not the Beowulf-poet and his ilk: they were above all the Latin poets that they studied in school. Chaucer himself lists them off in the envoy to Troilus – his greatest work, in his own estimation and mine, and the one on which he staked his reputation:
Go, litel book, go litel myn tragedie,Virgil, Ovid, Homer, Lucan, and Statius. Now, if you've had a modern literary education, you go along just fine with Virgil, Ovid, and Homer. Sure. We read 'em too. But at the end of the line we get a little shaky. Lucan? Maybe we've heard of him. Wrote about the civil war between Julius Caesar and Pompey, right? Fell afoul of Nero? Right, got it. Not that we've read him, but at least we have him placed. But Stace? Who's Stace? Someone who was just brought in because his name rhymed with pace?
Ther god thy maker yet, er that he dye,
So sende might to make in som comedie!
But litel book, no making thou nenvye,
But subgit be to alle poesye;
And kis the steppes, wher-as thou seest pace
Virgile, Ovyde, Omer, Lucan, and Stace.
No, Stace in fact is Statius, who wrote the Latin silver-age epic, the Thebaid. Chaucer refers to him a lot, in fact; he clearly read him closely and thought highly of him. We have never heard of him, and rightly so: his poetic merits are meager. But he was in Chaucer's pantheon of poets. Chaucer was daring to hope to rank with him.
This passage is just as interesting for the people it doesn't mention. (You will notice not a peep about any balladeer or Old English poet). Chaucer was fluent in French as well as Latin. He translated a fair amount of French poetry, and in fact adopted most of his English verse-forms from it. He may or may not have known Italian, but he knew either the originals or translations of Dante and Boccaccio: he probably borrowed more from Boccaccio than from any other writer. But he declines to mention any of them. He knows them, he's willing to use them, but he has no intention of kissing their footsteps. Earlier on he's mentioned precisely one English poet – his friend and contemporary Gower. The two of them intend, together, to put English poetry on the map.*
For all the apparent humility of this stanza, the gist is not humble at all. His tragedy, he's asserting, is on the same staircase as the great poems of antiquity. Chaucer is well known for his charming self-deprecation, poking fun at his waistline and his bumbling in love, but as a poet he takes himself dead seriously. He knows how good he is. And he knows that he is creating a new national literature. If you read the Hous of Fame and the Prologue to the Legend of Good Women you can see him hammering it out. But this is really the clearest statement. He's written a great tragedy, and now he must write a great comedy. He was to spend the last years of his career taking running starts at a great comedy, groping this way and that in the Canterbury Tales, not quite grasping that it was precisely the mismatchedness, the out-of-control-ness, the centrifugal force of the Tales that would be his lasting contribution to world literature. No sooner did he mount a narrator than the narrator ran away with him: and those runaway narratives, those careening voices, are what we really read Chaucer for: the drunken Miller interrupting the Knight, the wife of Bath justifying her marital strategies, the Pardoner explaining his cons. Chaucer never finished it, and he even tried to take it back, but actually he'd succeeded beyond his dreams.
* Yes, I've read Langland. No, I don't think his poetry is important, either in itself or as part of the English tradition.