The teams were evenly matched in the opening exchanges and Brechin went close on 24 minutes when a shot from Stuart Callaghan went just wide.
Arbroath opened the scoring on 36 minutes when Greig Henslee ran from the halfway line, linked with John McGlashan and fired the ball over Craig Nelson from 25 yards.
Brechin equalised after 39 minutes when Steven Hampshire raced clear on to a throughball and lobbed Neil Inglis.
Brechin made it 2-1 on 42 minutes when David White headed home a cross from Charlie King.
Arbroath equalised right on half-time with a superb free-kick from Gavin Swankie which went in off the crossbar.
Arbroath went close in the 57th minute when a header from Ian Diack was just kept out by Nelson.
After 76 minutes Graham Gibson had the ball in the net for Brechin but was ruled offside.
On 83 minutes a Gibson lob just cleared the Arbroath crossbar.
With a minute of normal time left Steven Rennie put Arbroath 3-2 ahead with a close-range volley from a Swankie cross.
A minute into stoppage time Grant Johnson drove the ball into the box and Hampshire got the final touch to make it 3-3.
And four minutes into added time Brechin grabbed a dramatic winner when Callaghan poked the ball home from close range.