In recent weeks Rangers have seen hopes of a historic quadruple shatter following a defeat to Zenit Saint Petersburg in the Uefa Cup final and a failure to record success in the Scottish Premier League.
"It means a great deal at the end of the season," Smith told Sky Sports. "We have suffered a few disappointments in the last nine days or so.
"It was always going to be a difficult match for us and, as it turned out, it was.
"We took a two-goal lead, but all credit to Queen of the South they came out in the second half and got back on level terms.
"But we still came back with the winner. I'm delighted for everyone."
Smith also hailed the effort of his players as they ended a tiring campaign with their hands on silverware.
"There was a tremendous amount of effort over the last couple of months and it is great to see them rewarded," added Smith.
"It has been difficult, but we have handled it and we have managed to win two cups this season.
"Every team thinks they deserve to win. If we had lost it would have been a cruel end to the season, but we won."