Any hopes Cambridge City had of putting pressure on the out of form League One side were dashed when Smart headed the Dons in front after only five minutes.
The striker was preferred to top scorer Izale McLeod and quickly vindicated caretaker manager Jimmy Gilligan's decision when he expertly flicked home a header from a perfectly placed free-kick by Nick Rizzo.
The Dons showed the gulf between the sides during the second half but lived dangerously at times during the first when the Conference South team created enough chances to go in level at the break.
Dave Sadler struck a shot against the far post in the 21st minute and in the 35th it took a brilliant full-stretch save from Dons keeper David Martin to fingertip a Matt Langston header away for a corner.
It was a helter-skelter cup tie littered with fouls and seven bookings as referee Grant Hegley struggled to keep control.
Cambridge had one more chance in the 70th minute but Alex Fiddes was unable to get the touch he needed from close range.
Dons dominated midfield for much of the second half, forcing Cambridge to defend for long spells and almost got a second goal six minutes from time when substitute McLeod hit a blistering 20-yard drive against the foot of a post.