Stoke boss Michael O'Neill promised there is much more to come from his side after they secured their first win of the Championship season with a 1-0 victory at 10-man Preston.
The Potters have scored just once in their first three league games but O'Neill is confident they will be tough to stop if they become more clinical.
"There are aspects that we need to improve on," said the ex-Northern Ireland manager.
"We were wasteful in the second half and when you're against 10 men it is difficult at times because teams commit bodies forward and you have to deal with long balls.
"It was important for us to try and win the game on the back of last week's defeat [2-0 to Bristol City]. We have four points from three games now and if we can take three points next week it will have been a good start.
"The red card, I thought it was a poor challenge. He doesn't really need to make that challenge and you can't tackle anymore. They are the rules.
"It is hard to say he was in control and it was a red card.
"With Lee [Gregory], we have a number of strikers who all want to play and feel they deserve to play.
"Lee has been better in games away from home and he did well on Wednesday against Gillingham so he deserved to start. He works hard for the team and was one of our better players.
"Overall, aspects of our game need to improve, we need to take care of the ball more and be more clinical, but it is still early and it was a great three points."
Barkhuizen saw red for a rash tackle on Stoke left-back Morgan Fox, ruining a good start for the hosts who created plenty of chances.
Gregory tapped in from close range shortly before half-time, running in unmarked at the back post to convert Jon Obi Mikel's low cross.
However, the Potters missed a series of chances after the break, with Sam Clucas and Steven Fletcher particularly guilty of being wasteful when in good positions.
Preston cranked up the pressure in the search of an equaliser but they failed to seriously test Adam Davies and fell to 19th in the table, with just one point from three games.
Boss Alex Neil said: "I wasn't sure about the red card.
"I've watched it back and he is running full speed, goes to slow down and loses his footing and slides into the lad, it was almost a slide tackle.
"It is not meant, it was next to their dugout and they were up appealing, the lad was screaming and rolling around.
"I've been in that situation, you know you're more than likely going to see a red when that controversy surrounds one tackle. There will be grounds for an appeal.
"It changed the course of the game. We were getting crosses into the box, we were a good threat. Everything was set up how we hoped and we thought we could go on and win it."