The former Chesterfield defender was hurt following a poor challenge by Quakers midfielder Will Hatfield in the 41st minute of Friday night's encounter.
Smith, who produced a man-of-the-match display on his debut against Knaresborough Town earlier in the week, received treatment but was unable to continue, limping from the pitch to be replaced by Ryan Fallowfield shortly before the interval.
Town were already in the ascendancy by this point, Mark Beck having towered above the Darlington defence to meet Warren Burrell's 22nd-minute cross from the right and nodded in his second goal in as many games.
Beck and Brendan Kiernan went close to extending the home advantage before Burrell did make it 2-0 five minutes after the resumption.
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Lloyd Kerry dinked a ball over the Quakers' defence for Burrell, up from the back, and he controlled it neatly before planting an effort past ex-Harrogate stopper Chris Elliot.
Jarrett Rivers reduced the deficit on 66 minutes, beating James Belshaw with a strike from 20 yards out, yet although there were further chances for Beck to add to his tally, no more goals were forthcoming.