Mark Stoneman (61) and Keaton Jenning's opening partnership of 96 helped Durham reach 180-5 from their 20 overs.
Glos slumped to 6-2 and 61-6 but Jack Taylor (80) staged a fightback, scoring nearly half of his side's runs.
But when Taylor was run out, Durham deservedly joined Northamptonshire and Nottinghamshire in the semi-finals.
Relive Durham's win over Gloucestershire as it happened
Durham's victory was significantly aided by some smart fielding that led to four run outs, as seven of Gloucestershire's batsmen failed to reach double figures.
But for Taylor's resilient 80 from 41 balls, which included five sixes and seven fours, the visitors could have won by a huge margin.
Neither side have ever won the T20 Blast and, while Durham's bid for a first T20 crown goes on, Gloucestershire are now set for a trophyless 2016.
Glamorgan host Yorkshire in the fourth quarter-final on Thursday.
Durham produced a dominant 19-run win to reach the 2016 T20 Blast Finals Day by knocking out South Group winners Gloucestershire in the quarter-finals.