Saints played out a high-tempo goalless draw against the Toffees to prolong their unbeaten league run to six games.
"Everyone is behind the new coaching staff and the new gaffer, and tonight we showed that," Schneiderlin told Saints Player
after an industrious midfield showing.
"He said to us that he didn't want to change too many things tonight because he didn't want us to try to learn too many things too quickly.
"We put in a good shift and I think we could have won the game, so I'm sure we will win some more games if we keep playing like that.
"The gaffer told us that we had seven clear chances in the first half, so we could have scored.
"That's football – when you don't score, you are exposed to conceding but today we stuck in there.
"We had a good defensive shape about ourselves because, when we can't win a game, the main thing is not to lose.
"Now we have another nine days [until Saints' next fixture] so we are going to know how we want to play," continued the Frenchman.
"We are going to get to know each-other better in the next couple of weeks. and I'm sure the future is going to be bright."