The latter stages of the first half then saw Stockley spurn two gilt-edged opportunities for Exeter as the opening 45 minutes eventually finished goalless.
Lincoln came out flying at the start of the second period, Pym deflecting Green's close-range effort wide for a corner and the Grecians having to hold firm at the back.
On 49 minutes, Green forced yet another save from Pym's legs before Elliot Whitehouse's effort on the rebound went over the bar.
Lincoln then hit the woodwork through just before the hour, with Exeter countering through Jake Taylor's close-range effort, which was parried away by Allsop.
For all of the two sides' endeavour to claim a first-leg goal, the game finished goalless and the tie remains on a knife's edge ahead of Thursday's second leg at St James' Park.
LINCOLN CITY (4-3-3): Allsop; Wilson, Waterfall, Wharton (Long 90+3'), Eardley; Whitehouse (Pett 84'), Bostwick, Woodyard; Rhead, Green, Anderson (Palmer 84')
EXETER CITY (4-1-3-2): Pym; Sweeney, Storey, Moore-Taylor, Moxey; Tillson; James, Boateng (Simpson 74'), Harley (Archibald-Henville 90'); Taylor, Stockley