Ashley Hemmings struck twice and Sam Mantom also claimed his second of the season before Nile Ranger pulled one back late on for the below-par hosts.
Smith said: "I have watched Swindon play a couple of times and they are a very good team but we were magnificent to a man and the game plan and tactics worked.
"The performance on the whole was excellent and it could have been more, their goalkeeper was fantastic.
"I was so pleased with our football and our movement.
"The discipline in the team was really important and they kept their shape really well because we knew there would be times where they would have periods with the ball.
"My players are getting better all the time and I am blessed to have them."
Swindon started brightly and Ranger had an acrobatic overhead kick cleared off the line by Craig Westcarr early on.
But it was Walsall who took the lead after 17 minutes when Hemmings took advantage of some sloppy Swindon defending to rifle in from the edge of the box.
The visitors doubled their lead two minutes into the second half when Westcarr's initial shot was beaten away by Swindon goalkeeper Wes Foderingham and Mantom was on hand to slot home the rebound.
Hemmings added his second and Walsall's third 11 minutes from time when he skipped through the Swindon defence and slotted past Foderingham.
Ranger reduced the deficit with an 83rd-minute header but it was too little too late for Mark Cooper's side.