The Blues went into the second-leg needing to overcome a narrow first-leg deficit after defeat at the Riverside Stadium earlier this month, giving Championship side Boro real hope of knocking Chelsea out.
But it will be Mauricio Pochettino's side at Wembley on February 25, against either Liverpool or Fulham, as Jonny Howson netted an own goal and Enzo Fernandez, Axel Disasi and Palmer all got on the scoresheet before half-time.
It was a more low-key second-half, but Palmer added his second of the night and Noni Madueke scored shortly after. Morgan Rogers netted a late consolation, but it did little to ease the pain for Boro in what was a 6-2 aggregate defeat.
Michael Carrick's side made a relatively confident start to the match, looking comfortable on the ball and preventing any early Chelsea charge, but the Blues got themselves level in the tie with 15 minutes on the clock.
From there it felt inevitable that Chelsea would take control of the tie and so it proved, as Broja failed to make contact with an effort but that worked out perfectly for Fernandez, free at the back post to sweep the ball beyond Tom Glover.
The visitors played more than a small role in their own downfall as the match quickly got away from them. A loose pass was intercepted by Disasi on the halfway line and charged all the way up the pitch, getting his reward as Sterling picked him out in the box for a first-time finish.
And three became four before the break, as Boro fell apart. Playing out from the back again cost them, Palmer on this occasion pouncing and rolling the ball into the bottom corner after winning it back high up the pitch.
Boro were more secure after the break, though Chelsea eased off with the result no longer in doubt. Familiar failings brought about further punishment for the visitors though.
Conor Gallagher was played in down the left after the Blues once again won the ball back deep in the Boro half, and he pulled it back perfectly for Palmer to score with a first-time finish.
Four minutes later, Gallagher jumped on yet another loose pass, sliding in Noni Madueke to beat Glover with the left-footed strike as Rav van den Berg could only boot the ball into the roof of the net in his attempts to clear on the line.
The Boro fans were at least given something to cheer in the closing stages, after Matt Crooks' header was ruled out for offside. Rogers worked space for the shot on the edge of the area and bent a finish into the far corner brilliantly, as the Boro fans finished the match in fine voice.