Killian Keane almost spoiled the Stevenson Park party with a second-half hat-trick, but Dungannon survived to squeeze through to a first semi-final appearance.
In a thriller which produced a hat-full of tries, nine in all, the Tyrone men looked as if the pressure might get to them. But as so often in the past, David Humphreys' right boot proved the steadying influence.
He kicked 19 points, including two late penalties, to send his side through to the last four amid triumphant scenes.
"That was a tricky game, and to win such a close game showed a lot of character in our side," said skipper Paddy Johns. "So we'll take the positives along with us and work on those things that went wrong and go all out in the semi-final."
Jan Cunningham's opening try, along with two Humphreys penalties and a conversion, helped the home side to a 13-3 lead, but unforced errors were finally punished when winger Kevin O'Riordan dived over from a ruck at the posts, with Keane converting to add to his earlier penalty.
Full back Bryn Cunningham sprinted through for a second try to give Dungannon an 18-10 interval lead, but two Keane tries inside four minutes on the restart rocked Willie Anderson's men.
Fortunately for Dungannon, Keane's scores sandwiched a Ryan Constable try, helping the Tyrone men to keep their noses in front. But Garryowen were gaining an edge in forward power, and when second row Shane Leahy powered over for another try, just a point separated the sides, despite Keane's missed conversion.
The Munster outhalf compensated with his third try, but it was Dungannon who finished strongest, Bryn Cunningham scoring his second, and Humphreys keeping his cool to guide over two pressure penalties.
Scoring Sequence: 2 mins: Keane pen, 03; 4: J Cunningham try, Humphreys conv, 7-3; 25: Humphreys pen, 10-3; 36: O'Riordan try, Keane conv, 13-10; 40: B Cunningham try, 18-10; 44: Humphreys pen 21-10; 46: Keane try, conv, 21-17; 47: Constable try, Humphreys conv, 28-17; 50: Keane try, 28-22; 57: Leahy try, 28-27; 61: Cunningham try, 33-27; 71: Humphreys pen, 36-27; 75: Keane try, conv, 36-34; 78: Humphreys pen, 39-34.
Dungannon: B Cunningham; J Cunningham, R Constable, J Bell, T Howe; D Humphreys, S Bell; R Mackey, N Brady, J Fitzpatrick, P Johns, A Kearney, A Boyd, A Hughes, T McWhirter. Replacements: M Haslett for Hughes, G Leslie for Mackey.
Garryowen: D Crotty; S Mackay, C Foley, J Brooks, K O'Riordan; K Keane, S McCarthy; N Hartigan, P Humphreys, R Laffen, F Costello, R Leahy, C Varley, C Hartigan, J O'Sullivan. Replacements: A Guicken for McCarthy, A Purcell for Foley, J Giltenane for Varley.
Referee: T Redmond (Leinster).