The visitors had taken the lead in the all-Championship affair when Matt Derbyshire claimed a typical poacher's goal.
However, the Clarets always pose a threat with striker Ings in such blistering form and he struck in each half to take his tally to nine for the season, and four in this competition, to send Burnley through.
In choosing his side, Forest boss Billy Davies repeated the trick of the previous round by changing his entire XI.
Even so, his line-up contained several players who were regulars in the team which narrowly missed the play-offs last season and it was Forest who asked more of the early questions.
Dexter Blackstock was left all alone from a corner but opted to try and head back across rather than going for goal, and the Clarets were caught napping again as Simon Gillett hooked a volley just past the post.
They quickly gained the advantage in the 24th minute thanks in part to a terrific piece of skill from summer signing Jamie Paterson.
The winger turned Kieran Trippier inside-out on the touchline then drilled across the face of goal and, although Blackstock's flick was denied by goalkeeper Tom Heaton, former Blackburn striker Derbyshire was on hand to prod home.
The hosts almost restored parity instantly, Junior Stanislas forcing Dorus De Vries into a smart stop while Sam Vokes nodded just wide after beating the flying Forest stopper to the ball.
And Sean Dyche's men would level thanks to a moment of magic from Ings on the stroke of half-time.
Trippier's cross found the 21-year-old on the edge of the area where Ings still had plenty to do, yet he controlled the ball and span around in one motion to rifle a half-volley straight in to the top corner.
Burnley almost grabbed a first-half stoppage-time lead too but Stanislas' drive was deflected wide and, from the resulting corner, Ings lashed over.
Dyche only fielded four changes from the weekend win over Leeds, however fatigue was certainly not an issue as De Vries made two fine saves to deny Stanislas and then Keith Treacy.
At the other end Jamaal Lascelles almost capitalised on a flicked-on free-kick before Derbyshire let Trippier off the hook in the 67th minute when he scooped over after the full-back's poor header back.
A minute later, Ings would make him rue the costly miss.
The ex-Bournemouth forward made a near-post run to meet Stanislas' set-piece and brilliantly flick a header over De Vries.
Davies introduced Ishmael Miller and he immediately posed a threat, heading wide then forcing Heaton to push over one-handed.
Ultimately, though, it would be Ings' double salvo, coming exactly a week after his brace against Birmingham, which would settle the tie.