Kirkland, a free signing from Wigan earlier this summer, pulled off a top-class save to deny Doncaster's summer signing David Syers from giving the home side a first-half lead.
Owls trialist Marlon Harewood appeared for the first half and other summer signings Nejc Pecnik and Chris Maguire were also on show for Dave Jones' Championship side.
Wednesday displayed a greater urgency to attack, with Pecnik seeing his first-time strike cleared off the line by Paul Quinn.
Rovers striker Chris Brown went close to snatching victory for his side in the 80th minute but lashed a shot over the crossbar when under pressure from Owls defender Miguel Llera.
