How to use Services to create new node with Entity Reference field?
I figured out the solution from comments on this issue.

To set the field via services, you can set the field's widget type to "autocomplete" and instead of just sending the nid, send a string with the title of the referenced node, a space, and the nid inside parenthesis.

'field_drawing' = {
    'und' = [
        0 = {
            'target_id' = 'Title of Node (123)'
        }
    ]
}