Two second-half goals from Pedro Neto did the damage for Xabi Alonso's visitors, who saw Emiliano Martinez stretched off late on, following a nasty collision with Jarrad Branthwaite.
In total, Tierney handed out seven bookings, the first of which was to Everton midfielder Harrison Armstrong for a crunching challenge on Josh Acheampong, but he opted against a second booking for Morgan Rogers, who caught James Garner with a late tackle.
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Moyes felt both calls could have been viewed differently, with Armstrong's punishment viewed as excessive by fans from both sides.
"If our football in this country is not going to allow tackles like Harrison's, then I don’t know.
"We’ve got referees who we want to be the best in Europe, or think they’re the best, because they also let things go.
"I thought it looked a half decent tackle. Maybe somebody might tell me I’m wrong, or tell me differently, but I wasn’t sure it was the right decision.
"I was disappointed with the decision at the time. I had looked at the Rogers tackle on Jimmy Garner, which maybe some might have said something different.
"I think it might have been a second yellow, but it wasn’t the case."