Shelvey joined Ian Holloway's side on loan from Liverpool on Friday and was involved in the game's opening goal as City goalkeeper Dean Gerken - recalled after an error-strewn display from David James in Tuesday's loss to Reading - spilt his long-range shot to leave Gary Taylor-Fletcher with a simple finish.
However there was nothing Gerken could do about Shelvey's strike 20 minutes after half-time when the former Charlton midfielder powered home a 35-yard free-kick.
The Robins provided enough as an attacking force to make Blackpool work for the points and Seasiders keeper Matt Gilks produced a splendid pair of saves from Brett Pitman and Neil Kilkenny either side of half-time.
However the wheels came off in spectacular fashion for Keith Millen's side as Bogdanovic and fellow substitute Brett Ormerod took advantage of their flagging opponents in the closing minutes.
Shelvey was prominent in the opening exchanges and drilled a second minute free-kick into the wall before fizzing a drive from distance wide.
Albert Adomah provided a lively counter-attacking outlet for City and he cut in from the right to shoot over in the 20th minute.
Neither goalkeeper had a save to make in the first half hour, but when Gerken was called into action eight minutes before the interval it was not a moment he will look back on with any great fondness.
Shelvey unleashed another effort from range which Gerken failed to gather, leaving Taylor-Fletcher to chalk up his fourth of the season.
The visitors looked to hit back immediately and Adomah had the ball in the net only to be ruled offside.
Gilks then saved superbly after Pitman steered Kilkenny's towards the top corner.
Barry Ferguson should have doubled his side's advantage in the penultimate minute of the half but blazed over from 12 yards after more good work from Shelvey.
City boss Keith Millen shook things up at the break, with Martyn Woolford and Jordan Spence replacing Jamal Campbell-Ryce and Ryan McGivern.
Kevin Phillips found himself through on goal for Blackpool shortly after the restart, but dragged his shot wide, something he was almost made to regret as Gilks plunged to push away a Kilkenny effort.
But when Marvin Elliot fouled Alex Baptiste, Shelvey had the opportunity to cap his performance with the flourish it deserved.
Baptiste touched the dead ball towards him and a fierce low piledriver left Gerken helpless.
Woolford might have reduced the arrears but slashed horribly wide, and City's seventh game without a win was about to get considerably worse.
First Ormerod crossed for Bogdanovic to slot home in the 83rd minute, and the veteran frontman was on target himself when Craig Cathcart knocked down Ferguson's cross from Shelvey's short corner.
That goal came in the second minute of stoppage time, but there was still time for Bogdanovic to add further misery for Millen's men.
Thomas Ince, the third member an inspired triple change from Holloway, provided the cross to leave Bogdanovic with another regulation finish.