The 36-year-old eventually prevailed in the 10th leg of the deciding set after his opponent had squandered three darts to win the match.
Salmon made a terrific start to his debut in the sport’s richest tournament and took the first two sets.
But Smith then started to show his form and hit back to level it up at 2-2 before his greater experience told as the tension mounted.
Next in line for the world number 21 will be UK Open runner-up Mark Walsh, who did not need to be anywhere near his best to account for New Zealand’s Warren Parry without dropping a set.
Seeded Darren Webster relinquished complete control against Holland’s Erwin Extercatte.
The 24-year-old timed his run to perfection, never leading until the final set and closing it out after Webster had missed two match darts.
Extercatte will meet world number one Colin Lloyd next after Lloyd defeated Gary Welding last night.
“I beat Colin 3-2 the only time I have played him in Holland and it was a really good game,” said Extercatte. “My chances would be small against him over here, but if I get an opportunity to win again I hope I can take it.”
Another Dutchman, Roland Scholten is also through after beating the legendary Dave Whitcombe 3-1 last night.