Having gone into the match on the back of four successive Championship wins, Wrexham had plenty of cause for optimism against a Forest side that showed eight changes from their morale-boosting 2-1 victory at West Ham United three days earlier.
After a low key start, Forest thought they had gone ahead on 18 minutes as Douglas Luiz drilled beyond Okonkwo from close range, only for the goal to eventually be chalked off due to a handball by the Brazilian midfielder.
The contest had been sparked into life though as Rathbone had a shot blocked shortly before Okonkwo parried Luiz’s drilled free-kick to safety at the other end.
But shortly after Sam Smith dragged a shot wide, Wrexham took the lead in superb fashion.
George Dobson robbed Omari Hutchinson of possession, and Cacace – on his first start in more than two months – rifled beyond Mats Sels from 20 yards out to register his first goal for the club in style.
They were in dreamland just three minutes later as Broadhead picked out Rathbone who was the beneficiary of a fortunate ricochet off the legs of Morata, and he coolly slotted beyond Sels to make it 2-0 with his fourth goal in eight outings since returning from injury.
The home faithful were only too happy to revel in Forest’s misfortune as much as their own joy, with fans aiming chants of “Premier League, you’re having a laugh” at Dyche’s side.
Rathbone almost turned provider in first half added time as he played Smith through on goal, although the striker’s shot was superbly smothered by Sels to ensure the top-flight side’s misery was not worsened.
Nicolas Dominguez, Morgan Gibbs-White and Neco Williams were all summoned by Dyche for the second half, much to the concern of those in red and white.
The visitors subsequently dominated proceedings after the restart as Luiz forced Okonkwo into a smart save from a free-kick.
Wrexham continued to carry a threat on the counter attack though, as evidenced when Sels had to be alert to keep out Broadhead’s close range shot.
Forest gave themselves hope on 64 minutes as Jesus nodded Nicolo Savona’s cross over the line, despite the best efforts of defender Callum Doyle.
A significant upset was back on the cards just 10 minutes later though as Dobson’s sumptuous delivery was met by Hyam whose bullet header flew beyond Sels – the defender’s first goal since joining from Blackburn Rovers on deadline day.
But Hudson-Odoi’s composed finish from 15 yards out just two minutes later got Forest back within a goal again to set-up a nervy finish.
And the 25-year-old showed great composure a minute from time to take the ball down on his chest before firing beyond Okonkwo to send the tie into extra-time.
Forest looked the likelier to net a winner, with Williams and Hutchinson testing Okonkwo after Taiwo Awoniyi and Dan Scarr headed efforts wide as the tie went to spot-kicks.
Josh Windass, Doyle, George Thomason and Jay Rodriguez were all on target for the Welsh side who celebrated wildly as Okonkwo produced two vital saves to see Wrexham through into round four in dramatic fashion.