An unfortunate own goal from Imps keeper Paul Farman had put the home side in front early on but former Stag Adam Smith equalised before the break before Briscoe won it with 13 minutes remaining.
After a 3-3 draw the first time around there were always going to be goals at Field Mill and in the 14th minute the first duly arrived, John Dempster's looping header cannoning back off the post and over the line off the helpless Farman.
Farman more than redeemed himself on the half-hour with a fantastic double save to deny first Matt Green and then Ben Hutchinson and just before half-time the away side were level.
It was their first real chance of the match and Smith, who only signed for Lincoln from Mansfield in the summer, broke the offside trap and coolly lobbed the ball over the onrushing Alan Marriott.
That set up a tense second half, Jamie Taylor coming close for the Imps and Dempster a constant menace at set-pieces for the hosts, before Briscoe side-footed home to set up a clash with the seven-time FA Cup winners in the New Year.
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