Adding the button to the UpdatePanel’s trigger tag, I got it working:
<asp:UpdatePanel ID="UpdatePanel" runat="server">
<ContentTemplate>
<asp:FileUpload ID="FileUpload" runat="server" />
<asp:Button ID="btnUpload" runat="server" Text="Upload"
OnClick = "btnUpLoad_OnClick" />
</ContentTemplate>
<Triggers>
<asp:PostBackTrigger ControlID = "btnUpload" />
</Triggers>
</asp:UpdatePanel>
I did not have to do anything different server-side (like user5159158’s answer).