The Saints sacked Adkins late last week and replaced him with Mauricio Pochettino from Espanyol but Moyes feels it was the former's legacy which allowed the south coast club to trouble his Toffees.
"After the way we played in the first half I was frustrated," he told Sky Sports 1. "In the second half we got good opportunities but we couldn't nick a goal.
"We didn't play well and Southampton have been playing well. I've been to see them a couple of times and they continued that form, particularly in the first half.
"I've got to say I thought their team did really well. I give Nigel (Adkins) a lot of credit. I saw his team play at Chelsea and they were excellent. Now there's a new manager taking over the reins and they've played well."