Latin does have diphthongs. Right off the top of my head:
ae as in caetera, Caesar
au as in haud
oe as in coelum
ua as in suaviter, ue as in Suetonious, but the 'u' is more like 'w', and partakes of the nature of a consonant
ua, ue, ui, uo as in any word after 'q', but this maybe a special case
ia, ie, io, iu but another special case, as the 'i' = 'j' and is more a consonant than a vowel.