The result capped a miserable week for the visitors, who failed to make amends for their shock Active Nation Scottish Cup exit.
Jamie Langfield, Sone Aluko and Jerel Ifil were the Aberdeen fall guys following Tuesday night's fifth-round replay defeat by Raith.
Manager Mark McGhee handed a debut to goalkeeper Stuart Nelson, as well as recalling Mulgrew and Fraser Fyvie in a bid to shake up the his underperforming side.
Opposite number Pressley also made three changes for his first home game since succeeding Eddie May, drafting in Showunmi and Moutinho up front and the fit-again Vitor Lima in midfield.
Today's game was the second meeting between these two sides this month, with the rock-bottom Bairns winning at Pittodrie for the first time in 52 years two-and-a-half weeks ago.
That was the last of their three SPL victories this season, while the Dons had not won since either.
McGhee had demanded a response from his side to the Raith defeat, which saw him spat at by furious fans as he left the field.
The fired-up Dons delivered inside five minutes today courtesy of a trademark Mulgrew free-kick.
Burton O'Brien was adjudged to have fouled Fyvie 20 yards out and dead-ball expert Mulgrew curled in a low effort Robert Olejnik got a hand on but failed to keep out.
The home side took less than eight minutes to hit back, Showunmi afforded too much time and space down the left before tricking Mulgrew and crossing low for Moutinho to sidefoot home via the far post.
Dons forward Darren Mackie immediately fired over from 20 yards before Lima made a hash of a sitter three minutes later, scuffing Showunmi's knock-down straight at Nelson.
Lima was pulling the strings for Falkirk and a delightful ball midway through the half set Ryan Flynn free but his finish was blocked into the arms of Nelson.
Jim Paterson had to be helped from the field in th 25th minute following a strong Showunmi challenge and he was quickly replaced by Davide Grassi.
Moutinho fired over 13 minutes before half-time after being found just inside the box by Marc Twaddle's low cross and the same player was off target with a rising volley three minutes later.
Aberdeen enjoyed a brief flurry of possession but Falkirk took a deserved half-time lead after winning a 42nd-minute free-kick.
Richard Foster was booked for a cynical foul on Showunmi on the left edge of the box and Scott Arfield crossed the ball in for Darren Barr to nod home his first goal of the season at the far post.
Showunmi could have made it 3-1 in stoppage-time but Mulgrew's shoulder challenge was enough to force him to shoot straight at Nelson.
Fyvie became the second Dons player booked after being adjudged to have felled Flynn seven minutes into the second half.
The visitors continued to make little impression, so McGhee threw on Alufor Michael Paton four minutes later.
Almost two years to the day since his last competitive game, Nelson was almost caught out by a 20-yard Arfield Drive on the hour mark, the ball hitting his chest and just evading the waiting Showunmi.
Aberdeen made their final change two minutes later, Paul Marshall coming on for Gary McDonald.
But despite another brief spell of possession, they were undone again by their own indiscipline 20 minutes from time.
O'Brien's free-kick found Flynn down the left and the young winger's cross was met by a sublime scissor kick from Showunmi, whose first goal in Scottish football capped a stellar individual display from the striker.
Marshall's first-time strike was straight down Olejnik's throat in stoppage-time, while Falkirk brought on Jack Compton for Flynn as they saw out what was arguably their finest performance of the season.
Source: Team Talk