England's 16-12 win over Robinson's men in Auckland and Argentina's 25-7 defeat of Georgia confirmed Scotland's third place in Pool B, meaning they have failed to reach the World Cup quarter-finals for the first time.
But Robinson, who earlier this year signed a contract extension until after the 2015 World Cup in England and believes he can turn Scotland into the northern hemisphere's leading nation, has Dodson's backing.
Speaking in Auckland, Dodson said: "We're frustrated and disappointed with the result against England, but clearly we're looking forward to learning from the experiences we've had while we've been out in New Zealand.
"We shouldn't get too carried away about what happened here. It was a loss, it was disappointing.
"What we have to do is to learn from this and make ourselves even more competitive and turn ourselves into a winning team.
"Andy knows full well what went wrong over here. He will go back, talk to his coaching group and I think he's going to look at the players and the ways in which we can be more successful going forward.
"I'm not in the business of losing; there's no way you can take comfort from a loss like this.
"We have to learn from it, but we don't have to accept it.
"We're going to build a winning culture within Scottish rugby.
"Andy Robinson's going to be a continuing part of that and I want the whole nation to remain buoyant about what we're going to achieve here."
Following all tournaments, a strategic performance review is conducted.
Robinson and his management group will be given time to reflect on the World Cup campaign, which delivered wins over Romania and Georgia and two narrow losses to Argentina and England.
Dodson added: "We'll listen to Andy and listen to the learnings that have taken place over here.
"Once a result like that happens there is a natural pause that takes place and a period of reflection and I don't want to rush Andy with this.
"He will want to tell us about what happened out here and he'll want to tell us what he's going to do to put it right.
"We'll be listening intently to that and giving him all the support he can possibly ask for."
Dodson is also keen to address issues of participation, youth development, finance and the Edinburgh and Glasgow pro teams in order to facilitate the improvement of the national team.
He added: "We still have a very clear road in mind about where the game's got to go, whatever happened in this tournament.
"Our plans won't be affected at all by what's happened.
"We know what we have to do with the pro game. We know what we have to do with community rugby and grassroots. We also know what we have to do with the national team.
"When we come back we'll put all those factors into place and build for the future."
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