Darwin Nunez had already struck the post with a magnificent curling drive by the time Curtis Jones nodded the hosts in front from full debutant James McConnell's cross.
Anthony Gordon, who returned to the Newcastle starting line-up after suspension, was a constant threat and came close to breaking the deadlock with a 20-yard drive that whistled inches wide.
The visitors pressed home their advantage after the break as former Foxes midfielder N'Golo Kante, gifted time and space space, drilled home a drive from 30 yards.
It was a tight and tense first half with Jordan Pickford making decent saves to deny Enzo Fernandez's rasping shot and Cole Palmer's dipping drive from range.
Southampton went close with Nathan Redmond's low drive from distance while Charlie Austin hit the post. It means Chelsea manager Antonio Conte will come up against managerial adversary Jose Mourinho at Wembley on 19 May.
Phil Foden sealed it, adding to the late goal he scored at the Etihad to give his side their first-leg lead with a fierce drive that found the Dortmund goal via the post.
Dallas gave Marcelo Bielsa’s side the lead against the run of play when he fired a low 18-yard drive in off the post shortly before half-time, but the game appeared to swing back in City’s favour when Liam Cooper was dismissed for a reckless
The Belgian settled a closely contested encounter with a spectacular right-foot drive that flew beyond Chelsea goalkeeper Kepa Arrizabalaga into the top corner from 25 yards in the 63rd minute.
A sensational long-range drive from Vinicius Jr gave Madrid an early lead at the Bernabeu. Bellingham then scored his 19th goal of the season from a sublime Vinicius pass before converting from close range after half-time.
But that would deprive Burnley their due for a disciplined, energetic and ruthless display that saw them score three times before the break, with Sam Vokes netting twice - a header and a volley - either side of a lashing angled drive from Stephen
The impressive James Trafford was kept busy in the Burnley goal and had already smothered Harvey Elliott's low drive before making a stunning fingertip stop to deny Mohamed Salah.
And he saw Rasmus Hojlund drive United into a third-minute lead. But Ratcliffe's pre-match assertion that United "normally do well against Spurs" wasn't quite accurate.
Manchester United captain Fernandes followed up an excellent pinpoint 25-yard drive by hooking back the cross five minutes from time that allowed Hojlund to score his first goal since February.
Foden extended City's lead after 62 minutes when he fired low past Villa's deputy keeper Robin Olsen after good work by Rodri, then claimed his treble in style seven minutes later with a fine drive into the top corner from the edge of the area