The Imps had fought back from two goals down after Gilles Sunu and Jay Thomas produced classy finishes to put the Gunners in control.
An Aaron Brown free-kick reduced the deficit three minutes later, and City were level within 10 minutes as summer signing Chris Fagan curled home from 25 yards.
But a corner 90 seconds from time saw Luke Ayling head home to win the game for the visitors.
Boss Peter Jackson told the club's official website: "I'm really pleased with my players. They are young, keen, very enthusiastic and never give up."
Jackson also revealed that he hopes to complete the signing of a striker over the next few days, adding: "Our back four are as good as anybody on their day. It's going forward that we need more options."
Source: Team_Talk