Hereford maintained their unbeaten start to the season with their first victory, against 10-man Chester.
The visitors broke the deadlock on the stroke of half-time as Luke Graham headed beyond John Danby from a Rob Purdie corner.
Chester's Michael Kay was sent off four minutes after the break for a second bookable offence.
From the resulting free-kick, Smith scored from six yards to inflict Chester's second successive defeat.
VIEW FROM THE DRESSING ROOM
Chester FC manager Neil Young told BBC Radio Merseyside:
"They've come with a game plan - they play a lot of long balls and I don't think they really bothered us at all, without us creating a great deal. We need to work on committing more men into the box.
"We've been undone by a very silly goal and it was very similar at Barnet on Saturday. We haven't done our jobs and it's cost us.
"Once you go a goal down against teams as organised as Hereford are, you're going to struggle to get back in."
Hereford United boss Martin Foyle told BBC Hereford & Worcester:
"It's always nice to get the first three points on the table, a clean sheet and an away win, and it can only do our confidence the world of good.
"There's still a lot of passages of play that we probably let ourselves down in at times, but sometimes you've got to come away from home, take it and move on to the next game.
"The goal deflated them just before half-time - then the sending off deflated them but you've got to go for the jugular and I think towards the end we had a lot of chances and you've got to be ruthless, but I'll take the 2-0 for now."
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