Dallaglio admitted Wasps' record in Twickenham finals - they have secured three Premiership titles and the Heineken Cup at English rugby's HQ over the last three seasons - was a major factor.
"We were far from our best but we showed what champions are about today. When you start the season there are only three trophies available so to win one is very pleasing. We are in the Heineken Cup and have a trophy in the cabinet.
"We have a real chance in the next six weeks to up our game and come back here at end of May in the Premiership final. That is the motivation - to try and retain the trophy we have won for the last three years."
Injuries overshadowed the first half, with Simon Easterby carried off unconscious after smashing his head into the shoulder of Wasps full-back Mark Van Gisbergen. Easterby had swallowed his tongue - Scarlets winger Mark Jones put him in the recovery position before paramedics got to the scene - but recovered to watch the second half from the bench.
At that stage the Scarlets led 10-3 but Wasps used the opportunity to regroup and drew level at the interval after Jeremy Staunton picked off a pass from Clive Stuart-Smith and sent Voyce under the posts.
Dallaglio said: "You don't like to see players stretchered off at all and we hope both players come through without any problems. But injuries are part of the game and Simon's injury gave us time to think about what we were doing."
The Scarlets were left to rue a number of missed opportunities in the first half and could feel the game slipping away after the interval as the rain lashed down and Wasps took a stranglehold on proceedings.
"It was a lost opportunity. There were 17 points in that first half for us, and we ended up going in 10-10 at half-time," said director of rugby Gareth Jenkins. "Wasps scored a good try at the beginning of the second half, and the weather then became a major factor. It was nearly impossible to play 'catch-up' rugby.
"They turned the screw and in the end they deserved to win. But if we had scored 17 points in that first half I wonder whether Wasps could have played 'catch-up' rugby more effectively than us."