Michael Carrick's side fell behind to the League One Grecians before being pegged back to 2-2 in the second half as Emmanuel Latte Lath sealed their passage into the last eight from the penalty spot.
Ryan Trevitt produced a stunning strike to give Exeter the lead inside 15 minutes – a lead they would hold going into half-time.
Carrick's side started the second half fast and were level thanks to an even better strike from the impressive Morgan Rogers.
Boro hit the front before the hour when Sam Silvera curled in another excellent goal before Trevitt was at it again to equalise – his effort from distance clipping Paddy McNair on its way to goal.
But the Championship side saw off their League One counterparts when Rogers was pulled over in the box with Latte Lath converting.
And here’s how Boro rated:
1. Tom Glover - 6
Not a lot he could have done about either goal from Trevitt. A little shaky once or twice in possession but largely untroubled. (Photo by George Wood/Getty Images) Photo: George Wood
2. Anfernee Dijksteel - 6
Continued his return to the side and was steady enough before being subbed late on. (Photo by Stu Forster/Getty Images) Photo: Stu Forster
3. Dael Fry - 7
Went close with a volley in the first half. Some solid defending late on to help thwart any Exeter pressure. Picture by FRANK REID Photo: Frank Reid
4. Paddy McNair - 6
By and large did what he had to do in front of him to limit Exeter. Was a little unlucky to see Trevitt's strike deflect in off him. (Photo by Ashley Allen/Getty Images) Photo: Ashley Allen