Boro striker Francois Zoko headed a 79th-minute equaliser to cancel out Nicky Ajose's 23rd-minute opener and take the game to penalties.
The visitors had a strong claim for a penalty in normal time after Zoko was tripped by Yaser Kasim but their appeals were waved away.
Westley said: "It is always devastating to lose on penalties and I thought we showed good spirit to come back away from home after going a goal behind.
"Swindon are on a good run and we knew this would be a very difficult game for us.
"It would have been nice to extend our run in this competition but it wasn't to be and now we can concentrate on the league and FA Cup.
"It is all about how we bounce back from this disappointment."