Simon Walton also completed the game but United, who were watched from the crowd by Eirik Bakke, were forced to settle for a point after passing up a series of chances.
Kevin Smith hit the post in the opening minute following a poor clearance from City goalkeeper Colin Doyle, while Einarsson and Crainey both went close before half-time.
Einarsson, however, was forced to withdraw in the 59th minute and Birmingham could have got the full points late on when Andrew Barrowman hit the ball against the woodwork.
Trialist goalkeeper Ben Smith was named in United's line-up, while the final few seconds saw the introduction of Academy prospect Tomi Ameobi, brother of Newcastle United striker Shola.
Leeds: Smith, Madden, Crainey, Hird, Gregan, Walton, Einarsson (Hutton 59), Smith, Howson (Ameobi 89), Bayly.
Subs Unused: Wilberforce, Camfield, Serrant.
City: Doyle, Melchiot, Painter, Oji, Bruce, Howland, Wright, Alsop, Barrowman, Kuqi (Allen 72), Birley.
Subs Unused: Sadler, Cottrill, Hall, Motteram.