Technically...yes you can transit without a Visa.
However, if traveling on 2 separate tickets you could run into troubles with through checking your luggage, or getting your boarding pass in Saigon.
/ 01MAY09 / 1956 UTC
National U.S.A. (US) /Embarkation Thailand (TH)
Transit Vietnam (VN) /Destination U.S.A. (US)
Visited Thailand (TH)
ALSO CHECK DESTINATION INFORMATION BELOW
Vietnam (VN)
TWOV (Transit Without Visa):
Visa required, except for Those continuing their journey on
the same day and provided holding confirmed return or onward
tickets. Leaving the airport transit area is not allowed(SEE
NOTE 26798).
NOTE 26798: Certain international flights operated by
Vietnam Airlines For details, click here
Visa Issuance:
- On arrival: if required, a transit visa when arriving by
Vietnam Airlines only, provided holding a letter from
Vietnamese Immigration confirming that a visa will be issued
on arrival. Fee: USD 25.-.
Additional Information:
- All visitors must hold:
a. documents required for next destination; and
b. sufficient funds for the duration of their stay; and
c. return/onward ticket, when not requiring a visa to enter
Vietnam.