Scotland striker James McFadden fired Alex McLeish's side closer to Euro 2008 and hailed the 'incredible' confidence among the players.
The Everton striker has now hit three goals in his past three internationals for the Group B leaders.
He said: 'Going into the game today, we thought it was probably the hardest game that we've had because of the expectation levels.
'Obviously we knew we could get the three points, but the whole country was expecting us to get the three points and sometimes it's hard to live up to the expectations.
'But we knew what we could do and we went about it the right way.
'If I get chances then obviously I'll get confidence that I'll score goals. But the confidence in the squad to win games is incredible.
'It showed today again. We just keep winning and long may it continue.'
McFadden felt he should have had a first-half penalty when he was tripped in the penalty area - and claimed referee Pieter Vink had owned up to his error.
'Absolutely, but the referee's admitted his mistake to me in the second half and that's all you can ask for,' McFadden said. 'Thankfully it wasn't a big mistake and we didn't need it.'
Scotland right-back Alan Hutton was named man of the match, and played his part in the clinching goal.
It was his pass from the right which found McFadden, who took the ball well before drilling it home in the 68th minute to restore Scotland's two-goal cushion.
Early goals from Kenny Miller and Lee McCulloch had put Scotland on top, but Andriy Shevchenko's close-range half-volley gave Ukraine hope and they dominated possession for long spells.
Hutton was relieved to get through another qualifier with three points safely secured.
'It's very important,' he said. 'We knew today we needed a performance. Three points were necessary and we got them, so all the boys in there are over the moon.'
The defensive effort from Scotland was at times desperate.
'It was very hard,' said Hutton.
'They've got some great players in their team. We defended well from the front. Great credit to all the boys.'
The Rangers full-back also had a good claim for a first-half penalty but the victory meant he was not too upset to be denied.
With Georgia next for Scotland, on Wednesday, a place in the finals is closer to becoming a reality.
'We're going to go over there and expect a really tough game,' Hutton said.
'It'd be a shame to go over there and not get three points after how well we've done here, so we're going over there to win.'