Rashford was brought into the starting lineup as one of eight changes to face Belgium on Thursday night, replacing rested attacker Sterling.
Smith believes that "passive" Rashford squandered his big audition in Kaliningrad on Thursday night, with Sterling the better option to face Colombia in the last 16.
"There was quite a ground swell from supporters as much as anything for Marcus Rashford to get a shout [to start for England] and if he'd have put that chance away it would have been a very different situation," he told Sky Sports News. "However, I thought he was a little bit passive at times.
"He wasn't as positive as he might have been in certain situations where he had a chance to take on his man. He's another very young player, though, and that's why at this stage of the competition you go with players like Sterling, who has so much more experience under his belt.
"He's also got that first game out of the way now and he gradually built his confidence against Panama. I would have loved to have seen him on the score sheet but I definitely think he makes for a better partner with Harry Kane. He's a different sort of player and he's definitely okay with this level of football. I'd definitely put him in over Rashford at this moment."
Sterling played a direct part in two of England's goals in their 6-1 win against Panama last weekend.