Carsley, who will be in charge of two matches this month in hopes of continuing through the autumn, previously played for Ireland. It was Rice and Grealish, both players with Irish connections, who quickly secured an early lead for England that was never in doubt.
Rice played three games for Ireland before switching to the country of his birth, while Grealish played at under-age level for England's near neighbour, who were also playing under new manager Heimir Hallgrimsson for the first time.
Nothing less than a convincing win over a nation ranked 54 places below them would do for Carsley to boost his credentials for the full-time role and the former England Under-21 coach stuck with Southgate's stalwarts for his first outing.
The visitors, playing in the competition's second-tier for the first time, took the lead moments after Ireland's Sammie Szmodics forced Jordan Pickford into a smart save, with Anthony Gordon wasting a one-on-one before Rice stroked the ball soundly into the top corner.
The Arsenal midfielder, booed by the home fans throughout, opted not to celebrate his fourth England goal. Grealish showed no such restraint 15 minutes later when he tucked away a Rice cutback after some tidy English passing easily cut Ireland open.
England's 352 completed first-half passes to Ireland's 50-odd highlighted the gulf between the side and left Hallgrimsson in little doubt that there would be no repeat of his memorable Euro 2016 win over England while in charge of Iceland.
England took their foot off the accelerator in the second half and Ireland finally showed some signs of life on the hour with the lively Ipswich attacker Szmodics flashing a shot just wide.
That was as close as the hosts got and England, who gave late debuts to Angel Gomes and Morgan Gibbs-White, should have added at least a third goal while seeing the game out at a comfortable pace.