The Architect's claim that the Matrix has undergone multiple cycles is corroborated by multiple programs. For example, the Oracle:
Neo: Then why didn’t you tell me about the Architect? Why didn’t you tell me about Zion, the Ones before me – why didn’t you tell me the truth?
Oracle: Because it wasn’t time for you to know.
The Matrix Revolutions (transcript)
Of course, the Oracle is in league with the Architect (he's using the system she devised to keep the Matrix stable) so one could argue that they both might be lying. However, some of the programs who corroborate the Architect's claim oppose the Oracle's system of the One. For example, the Merovingian (who hates the Oracle) speaks of Neo's predecessors:
Mark my words, boy, and mark them well. I have survived your predecessors, and I will survive you!
The Matrix Reloaded (transcript)
The Trainman, who works for the Merovingian, mentions that Zion was destroyed before:
Trainman: 72 hours. 72 hours!
Niobe: What'd you say?
Trainman: That's exactly how long Zion lasted last time.
Enter the Matrix (quote source)
Even Morpheus (and by extension the rest of Zion) knows that they were "freed" by the previous One and that he would "return":
When the Matrix was first built, there was a man born inside who had the ability to change whatever he wanted, to remake the Matrix as he saw fit. It was he who freed the first of us, taught us the truth: ‘As long as the Matrix exists the human race will never be free.’ After he died the Oracle prophesized his return and that his coming would hail the destruction of the Matrix, end the war, bring freedom to our people.
The Matrix (transcript)
Morpheus did not know about the path of the One and that it was a machine system of control because he only knew about Neo's immediate predecessor One. But what he did know is consistent with the Architect's claims (and rules out Neo as the first One, since Morpheus knows the previous One died). Importantly, the source of Morpheus' knowledge about the previous One comes from the first people to be "freed" -- it's a source that's independent from the machines.
We don't know for sure that Neo was the sixth One (as opposed to merely the second or third or whatever), but there is plenty of corroborating evidence that there have been multiple Ones and that the Architect is telling the truth.