Full-back Adam set-up the winning goal as Championship side Millwall produced a fine 2-1 win against Premier League Aston Villa at the New Den on Friday.
Villa took an early lead through Darren Bent but Danny Shittu powered home a header for the equaliser before Adam's key role in the last minute.
Picking up the ball on the right, the England U21 international floated in a cross met by John Marquis, who struck the crossbar with his initial header before nodding the rebound into the empty net for a dramatic winner.
Adam and the Lions now face a trip to Conference side Luton Town, who caused one of the biggest upsets in the recent history of the competition by toppling Harry and Norwich at Carrow Road on Saturday.
Harry started up front for the Canaries and was unlucky in the first half as a goalbound effort was blocked by Greg Taylor.
Grant Holt replaced Harry at half-time but there was no stopping the Hatters, who struck the winner through Scott Rendell with 10 minutes remaining to become the first non-league team to topple top-flight opposition since Sutton beat Coventry in 1989.