Yann Gboho gave Coventry the lead in the 52nd minute, but Barry equalised four minutes later with a header before putting Everton ahead in the 61st minute. Dewsbury-Hall completed the comeback in the 81st minute, finishing after Barry's pass.
David Moyes was pleased with the result but admitted Coventry had made life difficult for his side.
"We got a performance and we got a result but I have to say Coventry made it really difficult for us. You can see why they have picked up some really good results. They won at Forest and beat Sunderland the other week.
"It was good but in the end we got the goals which mattered and when you win in the Premier League you take it because it is so hard to get the wins."
Moyes was particularly delighted with Barry, who reached 10 goals for Everton.
"Important that Thierno Barry got another couple of goals tonight which was huge for us. He can smile at me now and say I told you so and that's good. We always liked him and we want him to keep scoring.
"He's missed quite a few chances but tonight he got important goals for us. He's becoming an important piece for us and he's got 10 goals for us which is really good."
Coventry City manager Frank Lampard felt several decisions had gone against his side, including an alleged hair pull by James Tarkowski on Caleb Yirenkyi, a challenge on Taiwo Awoniyi and the circumstances surrounding Barry's second goal.
"The VAR ones, VAR looked at them. The hair pull on Caleb, Taiwo I think he gets contact on the edge of the box which is a penalty. Then the second goal for them comes off the player's hand.
"If you're going to check the angles, there is an angle where it flicks in off his hand. VAR looked at it and decided not. The last touch to score is off the hand, in my opinion."
Lampard acknowledged that officials had reviewed the incidents but remained unconvinced by the outcomes.
"I've seen a conclusive angle so I can disagree. I understand they are going to back themselves that's fine and it's hard to say it's not a hair pull. It's an in-between kind of hair pull, that's a funny area to be in so I don't get that one either."
Moyes had not seen the alleged hair pull himself, while he also disputed the suggestion that Barry had handled the ball.
"I've seen the goal, I didn't think it hit his hand. I've certainly not seen the hair pull, that's for sure. I think a hair pull nowadays only gets you a booking, doesn't it?"
Everton's reaction to going behind pleased Moyes, who felt his team responded strongly after conceding.
"We gave away a really poor goal but the reaction from there was really good."
Lampard, meanwhile, insisted there were positives despite the defeat.
"A lot of good stuff in the way we played which makes it really difficult to lose the game. Big decisions affected the game, clearly.
"We have to keep working, I'm seeing progression in the team. We have to work hard. There is a good spirit with us and we have to keep improving."