The defender was on hand on the stroke of half-time to convert David Dunn's corner.
Both sides made one change each from last week's endeavours. Chris Taylor replaced Lee Williamson and Luke Varney played instead of the injured Dexter Blackstock.
Rovers created the first chance when a long throw-in was cleared as far as Dunn 20 yards out and he showed great technique to control and volley, forcing Paddy Kenny into a smart low save.
Taylor was lively in the early stages and produced a neat flick to send Todd Kane bombing down the right after 16 minutes, but the defender's shot was repelled by Kenny.
Another long throw from Spurr half an hour in caused havoc in the Leeds area. The ball once again found Dunn outside the box but this time he could only drill wide.
For all of Blackburn's efforts, it was Leeds who carved out the best chance of the half when Varney headed on to Ross McCormack, whose delightful flick found the onrushing Danny Pugh with just the goalkeeper to beat but Jake Kean produced a wonderful save to divert the ball behind.
Taylor continued to impress and forced a fine save from Kenny after 42 minutes when he collected Josh King's cross 20 yards out and swivelled before firing a left-foot shot that the Leeds goalkeeper did well to get down to.
Blackburn's pressure finally told in first-half injury time when Spurr got on the end of Dunn's corner to prod home.
Josh King produced a ferocious volley after 63 minutes that Kenny palmed behind.
And from the resulting corner, the ball eventually found King, who crossed for Grant Hanley, but the big defender's header sailed over.
Leeds showed signs of life with 16 minutes to go when Rudolph Austin fashioned space for a shot in the area but lashed into the side netting.
They threatened again with four minutes remaining when McCormack cut in from the left and fired a shot wide, though the Leeds players were incensed that a corner was not given.