With Milan not in action until Sunday, Inter built on their crucial 1-0 win over Juventus last weekend with a comfortable success at San Siro.
"I know how hard we work, I know how much the boys care, I know that we have brought home a [Italian] Super Cup," coach Simone Inzaghi told DAZN.
"But I know that by coaching Inter there may be criticisms at times, which is right that they should be made, yet I have always had confidence. I know the path we are pursuing.
"There are seven to go, we are all there, we will have to fight in every game because the pitfalls in this Serie A are always around the corner."
Before last weekend's hard-fought win over Juve, Inter had won one of their previous seven league games, allowing Milan to pull clear at the top of the table.
However, after the leaders were held to a 0-0 draw by Bologna on Monday, Inter knew victory over Verona would put them within touching distance of their city rivals.
Barella picked the perfect moment to score his first since January on the end of a Ivan Perisic cross, who also turned provider for Dzeko to net his 13th Serie A goal of the season.
Verona did rally in the second half, with Kevin Lasagna arrowing a fierce strike just over the top, but Inter remained in control until the final whistle.