Simon Ferry opened the scoring after just 57 seconds when he lashed home a cross from Gary Roberts, before Matt Ritchie made sure of the points with a low drive after 62 minutes.
Walsall's best effort came from a Jamie Paterson free-kick which Swindon keeper Wes Foderingham turned round a post at full stretch.
"Swindon are as good a team as we have played against in terms of physicality and the quality of their players. They are right up there," Smith said.
"We showed a lot of endeavour but were on the wrong side of the result. Swindon have spent a lot of money but this football club (Walsall) have good people who want to work here.
"A lot of clubs out there go into administration and have transfer embargoes, but this club runs a steady ship and will continue to do so."