Kiely's save from Zheng Zhi proved to be decisive as Albion won 4-3 to earn a trip to Peterborough.
Albion looked to be cruising to a comfortable victory when Roman Bednar, with his ninth goal in 11 games, and James Morrison, on target for the second time in successive matches, fired Tony Mowbray's side into a 2-0 lead.
However, the battling Addicks forced the game into extra time thanks to a 65th minute strike from Darren Ambrose and a first goal for the club by substitute Chris Dickson in stoppage time.
Substitute Sherjill MacDonald should have put Albion in control again ten minutes into extra time when he met a Bednar cross only to see a combination of goalkeeper Nicky Weaver and Madjid Bougherra bundle away his shot.
Dickson could have won the tie in the 117th minute when he homed in on goal with just Kiely to beat only to see his shot saved.
Luke Varney got the shoot-out off to the worst possible start for Charlton - smashing the first penalty against the bar.
Chris Brunt then scored for Albion and Dickson replied for Charlton before Albion skipper Jonathan Greening saw his effort saved by Weaver.
Jerome Thomas and Chris Brunt were both successful with their attempts before Zhi was crucially denied by Kiely.
Morrison put Albion in control, Bougherra scored for Charlton before Bednar scored the match winning penalty.
Albion should have won in normal time even though there was a touch of fortune about Bednar's 14th minute opener as he really didn't know much about it.
Greening's right-wing cross was headed back towards goal by Felipe Teixeira and flew over the line when the ball bounced off the on-rushing Bednar's shin.
Albion were fortunate to take a lead into half-time as Izale McLeod was on a one man mission to find an equaliser, but he was twice denied by Kiely and also saw a powerful 26th minute drive deflected over the bar by Leon Barnett.
McLeod then set-up a golden opening for his strike partner Varney in first-half stoppage time.
Pedro Pele was dispossessed by McLeod and the former MK Dons marksman rolled the ball across the area into the path of Varney who only had Kiely to beat but somehow managed to lift the ball over the bar from six yards.
It looked to be a turning point for Charlton and their fate appeared to be sealed six minutes into the second-half.
Teixeira was the provider again, this time rolling a pass into the path of Morrison who drilled home a low drive from 20 yards.
That looked to be game over as far as Charlton were concerned but they gave themselves hope thanks to the determination of Ambrose.
He won the ball on the edge of the Albion area and his pass then bounced back off Kiely and straight into his path allowing him to score his third goal of the season from 14 yards.
Lloyd Sam was then inches away from forcing the game into extra time when his 75th minute drive whistled across the face of the goal.
It was a warning that Albion failed to heed and they paid the penalty in the first minute of stoppage time.
Zheng Zhi rolled the ball into the area and Sam flicked the ball into the path of Dickson who made no mistake from six yards.