Stuart McCall applauded City's efforts despite watching their Johnstone's Paint Trophy dreams crushed by Carlisle.
But he attacked Simon Ramsden's red card which left his side to play the whole second half with ten men.
City had their chances before Carlisle, who had taken the lead straight after Ramsden's 39th-minute dismissal, sealed a 3-0 victory.
Greg Abbott's men booked a Northern Area final against Leeds, who were 2-0 winners over Accrington at Elland Road.
McCall said: "Everyone deserves credit. I thought we played ever so well and showed a great spirit and I felt for them at the back end of the game because they'd given so much.
"It's hard work with ten against any team, especially against one from the division above you when they're at home.
"How we never scored at the start of the second half I'll never know. We had three or four kicked off the line and played some really good stuff."
Ramsden, who will miss Saturday's trip to Aldershot, was sent packing after a second booking for pulling back Carlisle winger Matty Robson.
McCall criticised referee Tony Bates for "spoiling the game" and said to the Telegraph & Argus: "Again we're talking about an official for all the wrong reasons. It was an embarrassing decision.
"I'm disappointed for the fans, who came in great numbers and it's been spoiled.
"Normally when you go 3-0 down, people will get themselves off home but they stayed and gave us terrific support.
"The lads appreciated that and I think the fans will have appreciated the effort we put in and the good football."
But former City hero Abbott had no sympathy for Ramsden and felt the official got it right.
The Carlisle boss said: "When you've been booked, you can't grab hold of people and he pulled him back again.
"He was running the risk of getting sent off. Stuart will be upset but that's the nature of the beast.
"I was furious with their player because he kept pulling Matty Robson back. Matty's a key player in what we do and we like him to get at full backs and he was pulling him back every time."
City winger Omar Daley came off the bench for the last 20 minutes - his first action after a ten-month lay-off.
McCall said: "It's great for him to get back out on a pitch. He's still a long way off and unfortunately the reserve game against Scunthorpe has been cancelled.
"Omar's always going to be a threat and until he gets back up to full speed, we'll use him off the bench."
