Unai Emery completely changed his starting XI from the side that beat Aston Villa 3-2 in the Premier League on Monday (AEST) but a youthful Gunners side dominated throughout at Emirates Stadium.
Tierney, playing in his first game since recovering from groin injury following his transfer from Celtic, was rarely tested at left-back as Arsenal's defence enjoyed a comfortable evening.
Forest, sixth in the Championship, offered little as an attacking force and its defence was breached 31 minutes in when 18-year-old Martinelli headed in his first goal for the club.
Rob Holding, making his return after nearly 10 months out with a knee problem, converted from a corner in the 71st minute to put the game beyond any doubt.
Joe Willock and the excellent Reiss Nelson got in on the act before Martinelli doubled his tally with the aid of a deflection in stoppage-time.