The Sky Blues, who headed into the match level at 1-1 from last week's opening tussle, scored twice either side of half time at Meadow Lane to get the job done.
A first home defeat in 10 matches for County brings the curtain down on their season, while Coventry will now go on to face Exeter City - winners against Lincoln City in the other semi-final - at Wembley on May 28.
Marc McNulty's controversial late penalty at the Ricoh Arena six days ago gave the visitors something to build on, and they were soon ahead in the tie for the first time six minutes into the return fixture.
County failed to deal with the first ball and Maxime Biamou impressively steered it into the bottom corner from eight yards out, rewarding his side for their fast start.
Coventry had not lost a match in the league all campaign after edging in front, but they were almost pegged back as Jon Stead skied a first-time volley over the crossbar.
The visitors were themselves looking threatening, though, and McNulty wasted a couple of chances to move his side further in front before the interval, hooking one shot over the bar and driving a low effort on target for Adam Collin to deal with.
Star man McNulty got his goal 38 minutes in, however, as he found some space 12 yards out and converted Liam Kelly's clever pass.
The Magpies were living off scraps as half time approached, but from nowhere a long throw was flicked on and Jorge Grant thumped it home to keep the tie alive.
Things threatened to boil over late in the first half, with the referee deciding to caution a player on either side, and neither were willing to give anything away when play resumed.
The chances kept coming, as Michael Doyle came close from a free kick and Stead became the latest to test Burge at the other end.
Coventry then edged themselves closer to Wembley in a frantic five-minute spell as, after Jonathan Forte had what appeared to be a legitimate goal ruled out for offside, Biamou converted on the counter.
County had further reason to be annoyed as the linesman missed a couple of offside infringements in the build-up and, following a couple of near misses, it was game over thanks to Tom Bayliss's 20-yard drive that deflected into the back of the net.
NOTTS COUNTY (4-4-2): Collin; Tootle, Duffy, Brisley, Jones; Alessandra (Ameobi 78'), Hewitt, Noble, Grant; Forte, Stead
COVENTRY CITY (4-4-2): Burge; Grimmer, Willis, Hyam, Stokes; Bayliss, Doyle, Kelly, Shipley; McNulty (Ponticelli 90'), Biamou (Clarke-Harris 73')