Defender Smith blasted home his second goal of the season past the outstanding Chester City keeper John Danby, who did not deserve to finish on the losing side.
Danby had pulled off a string of fine saves, including three from debutante Blair Sturrock to keep his side in the game.
The penalty was given when Dave Artell handled Lucas Jutkiewicz's downward header in the ninth minute.
Smith stepped up to blast the ball home into the top-right corner beyond the despairing dive of Danby.
Swindon striker Christian Roberts put a free-kick just over the bar from 22 yards after Paul Linwood was booked for a foul on Jutkiewicz just outside the area.
Smith fancied his chance for a second goal and hit a fierce shot that Danby did well to hold on to under pressure from Michael Timlin.
The game was all Swindon in the early stages as they launched wave-after-wave of attack on the Chester goal.
Swindon left-back Andy Nicholas was next to try his luck on 22 minutes, when he tried a shot from 25 yards that went just wide of the left-hand post.
Roberts made the first of Sturrock's chances with a deep cross from the right-hand side that Sturrock met with a right-foot shot and Danby did well to palm away.
Chester had their first threat in the 34th minute when Alan Kearney put over a left-wing cross but ex-Swindon Town trainee Alex Meechan just failed to connect.
Swindon defender Jamie Vincent almost got his side's second in first-half injury time when he met Sofiane Zaaboub's corner and his right-foot shot was tipped over the bar by Danby.
Swindon continued to pile on the pressure in the second half but the elusive second goal just would not come.
Danby denied Sturrock with two fine saves and made fine stops from Jutkiewicz and Zaaboub.
Chester could not manage one shot on target and Kevin Sandwith had their only worthwhile effort with a 20-yard shot that went five yards wide.
Despite this, Chester almost grabbed an undeserved equaliser when Jerel Ifil let in substitute Chris Holroyd. The big defender made up for his lapse with a superb last-ditch tackle to secure the win for his side.