The Hammers were rewarded for their good start with Carroll heading home after nine minutes. However, they failed to kick on and were punished with Christian Stuani's equaliser before the half hour mark, but Carroll regained their lead just before the break.
The second period failed to live up to the first, with half chances from Adama Traore and Aaron Cresswell. But the game was wrapped up in extra-time with Calleri's effort.
West Ham were confident after last Saturday's win, and started brightly on Teesside.
Their first chance came two minutes in, when Cresswell played a ball across the six-yard box but Michail Antonio somehow didn't turn it home.
They were rewarded for their excellent start, when they took the lead at the Riverside. It was simple enough as Manuel Lanzini played the ball in, and Carroll rose highest to power his header home.
Back in it
Aitor Karanka's side had failed to turn up in the opening proceedings, but managed to equalise in the 26th minute.
It was good build-up play when Traore released Calum Chambers down the wing. The loanee defender played a simple pass across the area, and Stuani was there to hit it back across Darren Randolph into the far corner.
Traore nearly turned scorer in the 37th minute, as he did excellently to wriggles his way to the byline. He hit it across the face of goal, and Sam Byram did well to turn it behind rather than into his own net.
Slaven Bilic's side did well despite conceding, and the visitors did well as they managed to edge back ahead just before the break.
Antonio received the ball just on the edge of the area, and let one rip which was parried by Victor Valdes. The Spaniard palmed it into the path of Carroll, who did well to react and turn it home from close-range.
All quiet on Teesside
The second period was quiet from the beginning, with very minimal chances throughout the second period.
Traore tried a shot from distance in hr 56th minute, with De Roon managing to get on the end of it but it was pushed out by Randolph.
The home side came very close to an equaliser in thr 64th minute, when Alvaro Negredo put it across the face of goal. It was lashed against the crossbar by Cresswell, and Winston Reid did well to chest it into the arms of Randolph on the rebound.
It seemed that the Hammers had the game wrapped up as the game entered extra-time, but made sure of it with the third goal in the third minute of added time.
Alarm bells rang for Boro as the Hammers were on the counter, as the tore through what was a nonexistent defence. The chance looked to have gone as the pass was heavy for Lanzini, but he managed to bring it back for Calleri with his effort deflecting into the net.