Sinclair broke the deadlock in the 83rd minute and Dobbie added a second two minutes later to give the hosts their fourth successive home win.
Swansea boss Brendan Rodgers gave a debut to rookie striker Frank Nouble, 18, after he signed on loan from West Ham on Friday.
It was the only change to the Swans' starting line-up, with Dobbie dropping to the bench.
Scunthorpe were unchanged from the side that recorded a thumping 4-0 victory at Sheffield United in the Championship on Tuesday.
Swansea went close in the fifth minute when Sinclair hit a right-footed shot from 18 yards which drew a smart save from Scunthorpe goalkeeper Joe Murphy.
Seven minutes later the Swans threatened again as Nouble engineered the space to hit a right-footed shot from 12 yards which flashed narrowly wide.
Moments later the hosts went closer still when an inswinging corner from Mark Gower was greeted by defender Ashley Williams, whose firm header crashed against the crossbar.
Swansea midfielder Darren Pratley then fired a 25-yard shot just wide as the hosts continued to build up a head of steam.
Scunthorpe, buoyed by their thumping win at Bramall Lane in midweek, gradually gained a foothold on proceedings.
In the 18th minute, Cliff Byrne drilled in a fierce right-footed shot from 20 yards which drew a fine block by Williams.
In the 34th minute, Swansea's Joe Allen fired over the crossbar from 25 yards and then Nathan Dyer hit an 18-yard shot wide.
Genuine chances were few but the visitors threatened on the stroke of half-time when Jonathan Forte drilled a left-footed shot wide from 12 yards out.
Sinclair went close for Swansea but Byrne threatened for Scunthorpe five minutes after the restart with another strike from distance.
Shortly after the hour mark, Swansea carved out an opening when Forte fouled home defender Alan Tate and Mark Gower flighted in a dangerous free-kick.
The ball fell to Sinclair, but he volleyed a left-footed shot wide from six yards out.
Nouble was withdrawn in the 65th minute and replaced by Dobbie, but Scunthorpe continued to hold firm.
That ended in the 83rd minute when Swansea made the breakthrough thanks to Sinclair, who connected with a cross from substitute David Cotterill and fired home right-footed from six yards.
And Dobbie made sure of victory with five minutes remaining when he collected a pass from Gower and drilled the ball into the bottom corner from 12 yards out.