With the sides level at 1-1 after extra time, the visitors scored all five of their spot kicks and goalkeeper David Martin kept out Darryl Duffy's effort.
It was the Dons who struck first in normal time, Dean Bowditch finding the bottom corner following a snaking run from Jay O'Shea a minute before the interval. It was the perfect response from the visitors, who had seen midfielder Stephen Gleeson sent off four minutes earlier for his challenge on Sam Deering.
The Robins came back into the game in the second half as Chris Zebroski forced a save from Martin within five minutes of the restart.
Kaid Mohamed crashed in a 30-yard equaliser on 58 minutes before tricky winger Jermaine McGlashan dragged a shot inches wide of the Dons' far post.
The visitors also had chances to settle the game in normal time but found Cheltenham keeper Scott Brown blocking their way as he twice saved well from debutant Lowe.