As expected, manager Dave Jones made a number of changes, including Riccy Scimeca in midfield, for his first league start since August last year.
The visitors had former Bluebird Paul Parry in their starting line-up.
Michael Chopra went close in the 10th minute after receiving from Jay Bothroyd but his low shot went narrowly wide.
Three minutes later, Jon Parkin forced a flying save from David Marshall, who tipped his goalbound shot over the bar.
Preston had the ball in the net from the corner but the referee had whistled for an infringement.
Scimeca was in the thick of things for City but up front but, despite Bothroyd winning most of the headers, Chopra was unable to convert a couple of chances that came his way.
The Bluebirds began to get on top as the half drew to a close but the half time whistle came with most of the play having taken place in midfield.
Preston went close to opening the scoring in the second minute of the second half when Kevin McNaughton cleared a Parkin header off the line following a Billy Jones corner.
Anthony Gerrard was forced to leave the field in the 50th minute after cutting his head when accidentally kicked by Mark Hudson.
The ten men battled on and when Bothroyd's header went to Burke, he fired into the top corner to give the Bluebirds the lead.
In the 56th minute, Mark Kennedy came on to replace Gerrard with McNaughton moving into central defence.
On the hour, Ross McCormack replaced Chopra up front as City went for a second goal but an injury to Joe Ledley caused his withdrawal three minutes later with Gavin Rae going into midfield.
A triple substitution by Preston came in the 69th minute but it was Bothroyd who moved into space and powered a drive that flew just wide with visiting keeper Andy Lonergan well beaten.
There were one or two anxious moments as five minutes play was added on but the Bluebirds held out for a hard-earned victory.