Arsenal were sacked from the Carabao Cup on Wednesday night after losing 3-1 to West Ham in the fourth round.
Ben White headed an own goal past Aaron Ramsdale playing his first club game since September before Mohammed Kudus and Jarrod Bowen secured the win.
Martin Odegaard scored a consolation for Arsenal deep into stoppage time.
Rice, a £105 million signing for Arsenal from West Ham in the summer, came on as a second-half substitute.
But he could not stop his former club reaching the quarter-finals.
West Ham took the lead from a corner in the 16th minute with Bowen swinging the ball in close to Ramsdale, before White nodded it past him.
Tomas Soucek had hold of Ramsdale’s shirt, but with no VAR at this stage of the League Cup, the complaints of the England international fell on deaf ears.
Arsenal dominated possession but failed to break through in the first half. Eddie Nketiah skewed over their best chance from eight yards following Oleksandr Zinchenko’s knock-down.
And they were punished five minutes after the break when Kudus beautifully trapped Nayef Aguerd’s 50-yard cross-field pass before finishing low through the legs of Gabriel and into the far bottom corner.
West Ham’s win was safe shortly before the hour when Bowen collected a poor defensive header from White and fired in a volley from the edge of the area, a deflection off Gabriel’s thigh wrongfooting Ramsdale.
Arsenal captain Odegaard slotted into the bottom corner with the final kick of the game, but the Gunners’ 30-year wait to win the League Cup goes on, while West Ham are into the last eight for the third successive season.