Knight and fellow new signing Tom Williams were both included at the start, while the Dons showed just one change from the side beaten by Southampton in the cup.
Kenny Jackett's men were looking for a goal after two scoreless draws and a victory that would take them back to the top.
They didn't have long to wait as Knight struck in the fifth minute. Andy Robinson crossed, Craig Morgan's clearance went straight to the former Brighton striker and he beat Matt Baker with a first time shot into the corner of the net.
Eight minutes later he struck again. Adebayo Akinfenwa slipped him through the middle and he threaded the ball through the legs of Baker for his second goal.
His hat-trick came in the 27th minute after the lightening break by the Swans. Robinson crossed left to Williams and his centre was met by Knight who headed into the corner of the net despite being the smallest player on the pitch.
Swansea should have extended their lead in the 50th minute when Akinfenwa sent Leon Britton into space but Baker came out quickly to block.
Dons' half-time substitute Nicky Rizzo had a free-kick turned behind by Willy Gueret in a rare assault on the home goal.
The introduction of Rizzo gave the Dons a bit more urgency and just past the hour he had Gueret sprawling to clutch a low drive.
In the 79th minute Danny Wilson's side were awarded a penalty but Izale McLeod's spot-kick was saved by Gueret.
Referee Scott Mathieson called for a retake and this time McLeod found the net.
Swansea were forced to defend for much of the remaining 10 minutes but they held firm for a deserved victory.