Boro fell behind after 12 minutes when Anfernee Dijksteel scored an own goal when he was pressured by Sky Blues striker Maxime Biamou.
Pivotally, the Teessiders drew level four minutes before half-time when Grant Hall headed home Paddy McNair’s free-kick to score his first goal for the club.
Substitute George Saville then converted the winning goal in the 87th minute after coming off the bench and converting Jonny Howson’s low cross.
Boro’s other substitutes also made a difference as Duncan Watmore, Yannick Bolasie and Marvin Johnson were introduced in the closing stages.
It was Bolasie’s first appearance for nearly a month following a hamstring injury, and the winger almost added a third goal in stoppage-time.
Here’s how the visitors fared after the big win:
1. Marcus Bettinelli - 6
Didn’t face a shot on target after the goal. 6 Photo: Alex Burstow
2. Anfernee Dijksteel - 6
Caught out for the own goal but recovered as Boro looked solid defensively. 6 Photo: George Wood
3. Grant Hall - 8
Boro’s rock at the back. Hardly put a foot wrong and scored a crucial equaliser before half-time which swung the momentum. 8 Photo: Stu Forster
4. Dael Fry - 7
Dominated during a physical battle as both sides went direct in the first half. 7 Photo: Getty Images