Following a tense start, the tie, which was level at 1-1 from the first leg, sprung to life when Ipswich defender Christophe Berra was sent off for handball on the line and Wes Hoolahan slotted in the resulting penalty after 49 minutes.
However, the 99th East Anglia derby was all square again on the hour when defender Tommy Smith converted from close range
Norwich, though, rallied to take the lead again through Nathan Redmond before Cameron Jerome rolled in a third to send the Norfolk side into a £120million prize showdown against Middlesbrough at Wembley on Bank Holiday Monday
"I was surprised to hear people would think it would not be that tense, with what we were playing for in the biggest ever local derby, which is what I was being told," said Neil, who has overseen a fine run of form after he joined the club from Scottish Premiership side Hamilton in January
"Everybody reminds us how good a squad we have got, and you could see the fans getting a bit agitated in the first half, but the message to the players was to keep calm and not listen to the supporters.
"When they went down to 10 men, Ipswich had nothing to lose, and we conceded a sloppy goal, which I was not best pleased with, but after that, it takes bottle because the pressure is then straight onto us with expectation to go and break them down.
"We showed good composure and quality to create chances and could have scored more."
Norwich supporters spilled onto the pitch at the final whistle, before police horses were deployed in front of the Ipswich fans.
Despite the hype which will now surround the Canaries' first visit to Wembley since winning the 1985 Milk Cup, Neil intends to keep a lid on things back at their training base in Colney.
"It is for the supporters, they are the ones who should be enjoying it when we win, if we lose they will not be best pleased, so we have to make sure we do our best, but we will not be getting carried away with anything, because it is half a job so far," said Neil.
"The boys will be off tomorrow, and we have already got our week structured, and mine will be considerably different from theirs
I will be up to all hours making sure we are fully prepped as we can be and not leave anything to chance."
Defeat left Ipswich facing another campaign outside the elite clubs of English football, the Suffolk club having been relegated from the top flight in 2002.
Town boss Mick McCarthy said: "I am immensely proud of them, they have been fantastic all season.
"They are low maintenance, hard-working and professional and we have shown today why we're in the play-offs.
"The game has hinged on two penalty decisions
I thought we should have had one in the first half when Teddy Bishop is brought down and the second one is a stone-wall (for Norwich).
"It is a sending-off and with 10 men we were unlikely to win."
McCarthy added: "Good luck to the two of them, Middlesbrough and Norwich.
"One of them has to win it and one of them is going to be as sick as a seaside donkey at Wembley, just like I feel now, and will be coming back and seeing my kipper (face) in the Championship."
Source : PA
Source: PA