Swailes struck just eight minutes after former Dons player Paul Butler had a penalty saved by Willy Gueret.
It was a much different Dons side that took the field, with captain Keith Andrews, Dean Lewington, and Drissa Diallo adding to the already missing Mark Wright and Aaron Wilbraham.
It meant John Miles made his full debut with Carl Regan, Jordan Hadfield and Jude Stirling also coming in.
And there didn't seem to be any hangover from Monday's historic win when Dons created two early chances.
For the first Colin Cameron's delicate pass put Jemal Johnson through on goal, but he delayed slightly and John Danby blocked well with his legs.
Moments later Cameron set him up again as he cut in well from the left but he sliced his shot miles wide.
In a lively opening Chester then had a great chance. Swailes completely missed a routine clearance and Ellison ran through on goal, but Gueret was quick off his line and he poked wide.
Then to complete an action-packed first ten minutes Johnson outpaced Laurence Wilson and cut inside well, but his left-foot shot was scuffed harmlessly wide.
Things quietened as Chester grew into the game and on 20 minutes had a goal harshly ruled out.
Ellison easily outpaced Alan Navarro after a long clearance and tucked home when clear, but the referee harshly ruled that he had held back the covering midfielder.
The visitors continued to improve and the dangerous Ellison cut in well and had deflected shot that went just wide of the post.
They brought that form into the second half and got their penalty on 49 minutes when Hadfield brought down Chris Holroyd. But the ever-impressive Gueret saved the penalty low to his left from Butler.
It gave the Dons a lift and they upped their game and they got the opener soon after. Swailes impressively volleyed home from 12 yards directly from Cameron's corner.
They should have wrapped up the game on 72 minutes when after a lightning counter-attack Johnson put Miles through one-on-one, he rounded Danby but the covering Wilson somehow blocked superbly on the line.
They had another great chance just after but Cameron volleyed Lloyd Dyer's early cross wide from just six yards.
And to complete a trio of misses, Johnson fired straight at Danby after a superb turn in the box put him through on goal.