Winnard's back-pass in the first minute of the second half let in Dean Richards, whose shot came back off a post for Tendayi Darikwa to drive in the rebound from 15 yards to give Chesterfield their best-ever start to a season.
Chesterfield had good chances to score before Winnard's error, but bottom-of-the-table Accrington were no push-overs and could have ended their nine-hour wait for a goal in the closing minutes.
But despite another defeat, Beattie refused to heap all the blame at Winnard's door.
He said: "As I've said to the players, I'm delighted with the performance and the game plan worked again but unfortunately you can't do any amount of training for errors like the one that happened.
"Dean's devastated, but apart from that, he played a magnificent game as I've told him.
"When we came off, Paul Cook (Chesterfield manager) said no team this season has caused them the problems that we did.
"We've come to Chesterfield, who are top of the table, and we are expected to get beat, but we've played some really good football and made them change their formation because we were causing them problems.
"I've said to the players it's disappointing, but I need them to be mentally strong because we are in a situation where we need to pull together and if they keep performing like that, let me tell you it will change, we will take one of those chances and we could be talking about a 1-0 victory as easily as we are about a 1-0 defeat.
"It's hard because they are only young men, but I feel like I'm doing a good job in picking them up. We're disappointed in the result, but the performance was a good one and we can't be too disheartened."
Chesterfield assistant manager Leam Richardson said: "Full respect to Accrington, as many teams will this season they came here and tried to do a job and it's up to us to be patient and break teams down.
"A win's a win, we are top of the league and are there to be shot at. The easy thing is to get there, the hardest thing is to stay there and this week we don't have to look at anyone else's results.
"We are always going to have to be patient here because teams will identify we are a good team.
"We pass the ball well and move off the lines well and they will get behind the ball, which Accrington did, but we feel we have the players who can unlock teams.
"The lads who came off the pitch, the lads who came on the pitch and the ones who didn't enter the pitch all conduct themselves fantastically and if you've got that, the longer you will be top of the league."