Nmecha opened the scoring with a near-post header just after the half-hour mark, before Palmer added a second with an impressive solo effort minutes later.
And in a mirror image of the opening 45, Nmecha’s composed finish from Adrian Bernabe’s cutback sealed the three points in the 86th minute, before Palmer calmly rounded off the scoring just before the 90-minute mark.
Enzo Maresca’s men were full value for the win, and will take away plenty of positives ahead of their clash with Arsenal on Saturday.
We began brightly and almost found ourselves ahead just after the five-minute mark when Bernabe and Nmecha won possession off the back of a high press before feeding Liam Delap.
But after racing in on goal, the forward’s effort was brilliantly blocked by Leicester centre-half Ben Nelson.
Minutes later the ball fell kindly for Tommy Doyle around the penalty spot after some more smart work from Nmecha and Bernabe down the left, but he was denied a late 19th birthday present by a timely block.
The chances kept on coming for City, with Palmer cutting in on his left and forcing a smart save from visiting goalkeeper Jakub Stolarczyk.
Nmecha will then feel like he should have opened the scoring after losing his marker in the box, but his side-footed effort was high and wide from a wicked Doyle corner.
We continued to dominate possession, with captain CJ Egan-Riley often drifting centrally to offer more freedom for Palmer down the right.
And the breakthrough finally came in the 37th minute down that flank.
Luke Mbete’s timely interception from a dangerous cross set up a swift counter, with Palmer racing into space behind the Leicester backline.
The winger worked his way to the by-line before dinking an inviting ball back in to the onrushing Nmecha, who powered home with his head.
And Palmer went from provider to scorer just five minutes later, jinking into the box from the right and finishing from close-range to put some daylight between the sides.
We started in a similarly lively fashion in the second half and could have gone three ahead when Bernabe was sent clean through by Delap, but Stolarczyk was quick off his line to block the winger’s effort.
Leicester were then reduced to ten men just after the 50-minute mark when winger Ali Reghba was given his marching orders for a late challenge on Nmecha.
The visitors shored up their defence after this setback, but City continued to knock on the door as we tried to add further gloss to the scoreline.
Delap in particular went close just after the hour mark, pressing tirelessly from the front to win possession in the box, but his effort was well saved.
And the forward was denied the goal his performance deserved once again when a cheeky lob over Stolarczyk came back off the bar, before Bernabe’s diving header on the rebound was blocked heroically on the line.
But that elusive third eventually arrived with stoppage time approaching when Nmecha made no mistake with a cool finish from Bernabe’s cutback.
And not to be outdone, Palmer added his second two minutes later from close-range to put the game out of sight.
How We Lined Up
Claudio Gomes sat in front of the defence, with Doyle and Nmecha given licence to get forward in a midfield three.
What It Means
The win sees us move up to fifth in the Premier League 2 table, two points off the top four.
Enzo Maresca’s men are back in Premier League 2 action next Saturday as they welcome Arsenal to the Academy Stadium. Kick off is at 13:00.