Coventry City took the lead through Yann Gboho shortly after half-time, but Barry scored twice before turning provider for Dewsbury-Hall, who sealed the points with a powerful finish in the 81st minute.
The result completed an impressive week for Everton, who have won all three of their matches, and Dewsbury-Hall believes the team's mentality has been central to their improvement.
"Very, very important. It has been a fantastic week. Three games, three wins. I'm so happy with how the lads managed to respond to going down tonight. I'm just really happy we managed to get the three points here at home, so happy, happy days."
The midfielder believes Everton have developed an ability to recover from difficult situations, pointing to their late equalisers earlier in the campaign as evidence of a stronger collective belief.
"I just think our mentality this year has gone from strength. You saw in the early games this season we scored a couple last minute equalisers. We never know when we are out of a game. Look, every time we have gone down this season, we have reacted in a positive manner.
"Today it was almost like their goal kicked us into gear - which shouldn't happen really, but it did - and I thought after they scored we just turned it up a notch. We were fantastic for the last 30 minutes. They couldn't deal with it."
Dewsbury-Hall also spoke warmly about his developing relationship with Barry, whose two goals took his tally to 10 for the season.
"I'm really, really enjoying it. I've got a really good hybrid role in this team, where the gaffer is giving me the freedom to pop up all over the pitch. I feel like I'm getting a really good relationship with Thierno now.
"I'm buzzing that he keeps on getting the goals because he deserves it so much. He's come back from the summer like a different player, and all credit to him because without him tonight - to kick-started us into the gear - we probably don't win the game."
Barry was equally pleased with his contribution, explaining that his work during the summer was beginning to pay off.
"It's always a pleasure to score and for my team to win so I'm very happy. I just work a lot during this summer so we can see the results so I'm happy and I need to continue like this to help my team win every game."
The striker admitted that missed chances can affect his confidence, but insisted he has continued working to improve.
"To be fair sometimes it is difficult when you miss a lot of chances your mind is sometimes bad but I always work to do my best."
There was also a personal disappointment for Dewsbury-Hall, who was left out of Thomas Tuchel's latest England squad despite his strong club form.
The midfielder admitted the decision was difficult to take, but said he would continue pushing for an international opportunity.
"It was a disappointing morning, I'm not going to lie. It's a massive dream to play for my country. I know I was close. Yeah, sad, but really proud of myself and how I have reacted.
"All I can keep doing is putting in good performances and hopefully one day I get the nod, because I'm never going to give up. I've got the mentality. I'll keep pushing and pushing and see what happens."
Barry, meanwhile, is enjoying life at Hill Dickinson Stadium and was keen to praise Everton's supporters.
"It's always a pleasure to play at home. The fans here are incredible. They sing from the first minute until the end. It is a real pleasure to score here and to play here."
The Frenchman also revealed the inspiration behind his celebration, explaining that he had borrowed the dance from an NFL video he had watched on TikTok the previous day. He had planned to perform it with Tyrique George if they scored, but ended up doing it alone.
"I was on TikTok yesterday and I saw an NFL guy do the celebration and I said to Tyrique if we score tomorrow we do the dance like that but I think he was shy to do it so I do the dance alone."