Smith said his squad had "a tremendous desire" to win games and they showed that at Hampden.
The Rangers manager said of his penalty shoot-out hero: "When it comes to the last penalty, there is no-one better to take it than Boyd."
An entertaining match finished 2-2 after extra time, with Boyd twice hauling his side back into the game after goals from Noel Hunt and Mark de Vries.
Goalkeeper Allan McGregor saved two spot-kicks in the shoot-out but was quick to heap praise on match-winner Boyd.
"It's an unbelievable achievement - all the boys have dug in so much it's untrue and our team spirit has been unbelievable - and that's what's come through in the end," said the delighted keeper.
"All through the game, I thought, 'it's not going to happen,' so to win on penalties was brilliant.
"I was feeling all right during the shoot-out and felt I could save a couple and luckily I did.
"Boydy hasn't been a regular player but he came in and I honestly think he won it for us - he's been brilliant and he's a good friend so I'm just really pleased for him personally."
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