Filipe Morais opened the scoring from the penalty spot in the 51st minute and Jordan Burrow added a second to secure a Championship scalp for the League One side.
This is the third season in a row that the Tractor Boys have been knocked out of this competition by lower-league opposition as two second-half goals sent the League One outfit through.
Christophe Berra came close to giving Ipswich an early lead, but his downward header from Paul Anderson's corner was cleared off the line by Sam Wedgbury.
Stevenage were to go ahead six minutes after the restart when Oumare Tounkara was clipped just inside the box by Luke Hyam and Morais confidently dispatched his penalty.
David McGoldrick almost pulled the visitors level three minutes after he was brought on when he was denied by a superb save by Boro goalkeeper Steve Arnold.
And the hosts were to double their lead and seal the game with 15 minutes left as Luke Freeman's low cross was turned in from close range by Burrow, for his his first goal for Stevenage.
More to follow