Leading 29-18 following last week's first leg at Headingley, Irish – who topped the regular season table – completed the job to win 84-66 on aggregate in one of the most entertaining games of the campaign.
In an exhilarating first 20 minutes, the sides traded tries as Canadian international Ciaran Hearn and scrum-half Brendan McKibbin's efforts for Irish were cancelled out by Seb Stegmann's clinical finish and Mike Mayhew's quick thinking.
Fergus Mulchrone also dived over for the hosts as they took a 13-point aggregate advantage into half-time before Alex Lewington's scintillating solo effort and McKibbin's second just after the break ended the tie as a contest.
Lewington dotted down again and Bell kept the scoreboard ticking over from the tee before Ben West, Sione Faletau, Stegmann's second and a penalty try bought Carnegie closer.
And although both sides had a man – Ben Franks and Charlie Beech – sent off, that could not spoil a spectacular evening as Irish earned promotion back to the top flight after last year's relegation.