Salisbury started brightly and more than held their own against their more illustrious opponents in the opening quarter of the match.
However, Forest slowly started to gain control and took the lead in the 27th minute when a corner beat everyone except Nathan Tyson who was lurking at the far post and he shot home from a tight angle.
Goalkeeper Ryan Clarke was in fine form for Salisbury and kept his side in the tie with a string of fine saves, and the home team fought their way back into the match after the interval.
Nicky Southall was forced to head Wayne Turk's volley off the line before Matt Tubbs showed great skill inside the area to make space for himself before firing home from close range.
If Salisbury can beat their League One opponents in the City Ground replay, they will play Premiership strugglers Charlton Athletic in the third round.