A 3-1 win against Coalville Town started the Minstermen's defence of the trophy they won last season, writes Tom Feaheny.
"We were good and strong first-half, we should have been three up, we didn't take some of those chances, but in the end it was all about winning the game," Gray told BBC Radio York.
"I liked Jon (Parkin) again getting on the scoresheet, he's in form, he wants to play and he is an outstanding player for us, James Gray had another good game.
"The goal we conceded was frustrating, it was another cheap goal we gave away, things like that need to change, we haven't kept a clean sheet since I came in, that has to be the aim.
"The lads have been fantastic since I came in, we are doing alright, but for the aims we have, we know we can improve."
For more news, reaction and action from across the National League, like us on Facebook and follow us on Twitter and Instagram.