A 16th-minute penalty from Sam Baldock was enough to beat an off-key Saddlers side, who rarely tested home goalkeeper Elliott Parish.
Smith said: "For some reason we weren't at it. We didn't use the ball well enough and Bristol created better chances than we did.
"The penalty decision was correct and, although I could moan about some of their time-wasting tactics, I can't have any complaints about the result.
"If we had been on our game we would have got something from the match, but that wasn't the case. Bristol worked hard and made things difficult for us, but we had an off day.
"We won't dwell on it and one thing I can say about my players is that when they have a setback they tend to bounce back quickly, so I shall be expecting a positive response in our next game at Gillingham."
The deadlock was broken in the 16th minute when City midfielder Bobby Reid was tripped by Saddlers goalkeeper Richard O'Donnell as he burst into the box on the left.
Baldock found the middle of the goal with a drilled spot-kick and might have gone on to score four or five.
Reid had to be replaced because of injury after 27 minutes, but the home side stayed in control and could have been out of sight by the break as Baldock squandered two great chances.
Walsall were restricted to long-range efforts in the first half and went close just before the interval when Romaine Sawyers fired narrowly wide from the edge of the box.
The visitors lifted their game in the second half, but were still unable to force a testing save from Parish.
At the other end Baldock had two more chances to make the game safe, but on each occasion was denied by goalkeeper O'Donnell.
- Sports & Recreation
- Sam Baldock