Burnley came within three minutes of knocking out Tottenham in midweek and booking a place in the Wembley final.
But they picked themselves off the floor to force a 2-2 draw at The Hawthorns thanks to a last-gasp equaliser from Martin Paterson.
Mowbray was frustrated at his side being held to another replay after Peterborough forced a 1-1 draw in the third round.
He said: "It is hugely frustrating but it is part and parcel of football and we have got to get on with it.
"I was disappointed with the performance in the first half. You have got to give Burnley a lot of credit. They play the game in a manner I appreciate.
"Having got in front at half-time, we should have gone on to finish the game off and we had enough opportunities after half-time to do that.
"At 3-1 it would have been a long way back for them after all the effort they put in this week.
"They always had something to chase and go for and they got a goal and good luck to them. They took a wee chance by pushing extra men forward and they reaped the rewards."
Skipper Jonathan Greening has been added to Albion's lengthy injury list after suffering a medial ligament problem.
Mowbray said: "Until we get a scan, we won't know how long exactly he will be out but I would suggest it will be at least two to two and a half weeks.
"Greening has been our most influential player and has been since I walked in the building. He said it was his first injury in 14 years and he is a bit distraught about it. Let's hope it is a short absence."