Sauron was the first to create Uruk-hai - they first appeared out of Mordor in the last years of Steward Denethor I, before TA 2475 (Appendix A). This was long before Saruman's treachery that roughly dated to the early 3000s TA.
An argument could be made, however, that the Uruk-hai in the LotR were a different breed created by Saruman by inter-breeding Orcs and Men. There is a quote from the History of Middle-Earth (Morgoth's Ring) that states
There is no doubt that long afterwards, in the Third Age, Saruman
rediscovered this, or learned of it in lore, and in his lust for
mastery committed this, his wickedest deed: the interbreeding of Orcs
and Men, producing both Men-orcs large and cunning, and Orc-men
treacherous and vile.
"This" being the creation of Orcs.
Treebeard also speculates on the Uruk-hai's creation in LotR, saying
For these Isengarders are more like wicked Men. It is a mark of evil
things that came in the Great Darkness that they cannot abide the Sun,
but Saruman's Orcs can endure it, even if they hate it. I wonder what
he has done? Are they Men he has ruined, or has he blended the races
of Orcs and Men? That would be a black evil!
That is in-character speculation rather than a definite statement though.