It took four minutes for Cardiff to take the lead, when a long ball found Nicky Maynard, and the striker's flick eventually fell to Burgstaller who finished inside the area.
The second half was much more of a contest, but Cardiff got the crucial second in the 81st minute when Haynes turned Mats Moller Daehli's cross into his own net.
The Sky Blues' own substitutes provided them with a glimmer of hope, when Miller finished Haynes' cross from close range.
And the hosts could have taken the game to extra time but Andy Webster's header fizzed just wide.
But that was the last action as the Championship side progressed to the next round.
Summer signing Marc McNulty scored a late winner for Sheffield United to see off Mansfield 2-1 in the first round of the League Cup.
McNulty sent the Blades through after Alex Russell levelled for Mansfield just four minutes after Paul Butler's opener.
United's Jose Baxter had an early shot beaten away by goalkeeper Sascha Studler while Neill Collins and Butler both headed off-target.
Mansfield threatened when Simon Heslop made a good run at the defence and laid the ball off to Sam Clucas whose low shot was saved by Mark Howard.
Heslop saw an effort deflected wide soon after the re-start and a header from Clucas flicked the top of the bar before Butler headed home at the far post in the 53rd minute after Davies' corner caused problems for Studler.
But Fisher drew the League Two side level with a shot from inside the area after Lee Beevers' effort had struck a defender.
The game looked to be heading for extra time until McNulty grabbed his first Blades goal in the 86th minute to give the home side victory. The forward held off Ritchie Sutton's challenge inside the area before firing hard and low past Studler.
- Sports & Recreation
- Ryan Haynes
- Guido Burgstaller
- Shaun Miller
- Paul Butler
- Nicky Maynard