In the first half, Allan Dempsie and Martin Harty both chanced their luck from range, but both missed the target. Fraser Bremner came closer after chasing Harty's through ball but his shot from the edge of the box went inches wide of the right hand post.
However, it was Albion who had the better chance to open the scoring. Ten yards from goal, Paul McManus found himself with the ball at his feet following a Marc McKenzie corner. Fortunately for Elgin the former Raith Rovers striker, who already has three goals to his name this season, shot wildly over.
City went ahead in the 72nd minute. Bremner tore down the right to provide the ammunition for a goalmouth scramble which ended with Nelson firing in from the back post.
The Borough Briggs side held onto their lead to secure the 1-0 win but they were given a late scare when Marc McKenzie hit a half volley off the far post in the dying minutes.
City: Kieron Renton, Stuart Cumming, Allan Dempsie, Paul Kaczan, Hugh Dickson, Jamie McKenzie, Fraser Bremner, Adam Nelson, Martin Harty, Willie Martin, Andy Roddie (Paul Napier, 82)
Subs not used: Jason McDonald, Garry Wood, Iain Vigurs, Martin Pirie
Goal: Adam Nelson, 72