The Light Blues reached their first European final in 36 years with a dramatic penalty shoot-out win over Fiorentina in Florence, and will face Dick Advocaat's Zenit St Petersburg at the City of Manchester Stadium on May 14.
Strachan said: "It is a fantastic achievement for the lads and Walter especially.
"I know Walter well and he is a friend.
"I am only happy for the coaching staff and the players - get that right - because I know how much hard work they have had to put in to get to this stage and they really should be proud of themselves.
"If we can't beat them in a competition then there is nothing we can do about it.
"We weren't in that competition and they have been very successful.
"And it's good for Scottish football, the profile is higher and it helps us to get further up the UEFA rankings."
Strachan was reluctant to reveal if he had sent any congratulatory texts or messages to Smith, saying: "If I have then it has been in private and no-one needs to know about it."