Spurs at one stage were 3-0 down before staging a fight-back.
Postecoglou said afterwards: "It is a tough result for us. In the first half we struggled to get control of the day. We made some changes that probably weren't helpful in the end. Players struggled to come to terms with it and Everton capitalised. The players put in a good response in the second half and credit to them but it wasn't enough.
"We had a great chance at 1-0 so we had opportunities to stay in the game. The first half we gave them too many opportunities to hurt us. The second half response was better.
"When you're 3-0 down it can one of two ways. Players can lose discipline and focus but they were determined to get something from the game. The players that came on helped us but sadly it wasn't enough.
"There is always encouragement or the players but ultimately we've fallen short and haven't won a game of football. We have to try and find a way out of it.
"The fans are justifiably disappointed. They are backing us and the majority understand the position we're in. we're missing a lot of players at the moment and that is making it difficult for us. Dominic Solanke picked up a knee injury in training yesterday and we don't know the extent of it yet."