The Terriers were looking to shake off last season's play-off final disappointment and winger Gary Roberts appeared to have set them on their way when he lashed home from close range in the 65th minute.
But Richie Barker's Bury, runners-up in League Two last time out, upset the odds when Lowe turned the ball home from close range in the 76th minute.
Defender Calum Woods made his first start for Huddersfield while fellow debutants Oscar Gobern, Donal McDermott and Anton Robinson formed a new-look midfield.
Bury's summer signings Mark Hughes and Mark Cullen both started but the latter was forced off injured in the 23rd minute and replaced by Andy Bishop.
Huddersfield went ahead when substitute Lee Novak's cross was converted by Roberts from close range.
But Town's lead did not last long. Bishop headed the ball on in a crowded penalty area and strike partner Lowe pounced to turn home a 76th-minute equaliser.