"In order to break that stalemate, to increase the capacity, the decision was made to bring many more forces into the south," he added.
The BBC's Martin Patience, in the Afghan capital Kabul, says Gen Dutton's comments may add to criticism that British forces have been overstretched and under-resourced.
This is denied by the MoD, but military officials admit that progress has been limited.
Gen Dutton, deputy commander of Nato's International Security Assistance Force (Isaf) in Afghanistan, said "one of the other reasons for optimism" was the widespread realisation that Pakistan was just as important to the war effort as Afghanistan.
He said the border was now "much better controlled" and would become increasingly so in future.
"Don't get me wrong here, I'm not saying we're now in a position where we wholly control the border and the border has become irrelevant - far from it.
"But we are in a much better position in that respect than we have been in the past."
Air Chief Marshal Sir Jock Stirrup, Chief of the Defence Staff, agreed that real progress was already being made with existing resources.
"No commander would ever say no to more forces, but the question is, can we actually deliver what's required in terms of improved governance in Helmand with the people who are here at the moment? The answer to that is quite clearly yes.
But he added: "With more resources, we could do more and we could do it faster.
"But you can see by going around at the moment that, where we've got our people on the ground providing security, real governance is starting to emerge very successfully here in Helmand."
General Sir Richard Dannatt, the head of the Army, said the increase in American troop numbers was "extremely welcome".
He added: "We're working hard to build up the Afghan National Army and, as I've said before, I don't really mind whose feet are in the boots on the ground as long as we've got the right number of boots on the ground.
"The Americans, the Afghans, the British, the Estonians, the Danes - we're all working together."
He also said the operation demonstrated "that when the time is right we'll move on and deal with an area where the Taliban are, so that we can then secure the population and begin the development in that area as well".
In a statement, the MoD said: "We welcome the deployment of additional US troops to Helmand and southern Afghanistan. These troops are being deployed where they are needed the most.
"British Forces have achieved a great deal and these additional resources will allow us to hold the areas that we have cleared and allow us to continue to set the conditions for improved governance and social and economic progress."
In March, President Obama said an extra 4,000 US troops would help train up the Afghan army and police.
This was in addition to the 17,000 troops whose redeployment to Afghanistan, mostly in the south of the country, was unveiled earlier in the year.
British troops have been on operations in Afghanistan since 2001
The UK has about 8,000 troops serving in Afghanistan, most of them in Helmand.
In April, British Prime Minister Gordon Brown offered to send more UK troops to provide security ahead of the country's presidential elections in August.
This would be a temporary deployment involving hundreds rather than thousands of troops, and conditional on other Nato members making a similar move.
Gen Dutton also said commanders were going to be "even more restrictive" on soldiers' freedoms to try to prevent civilian casualties.
"Things are still going to go wrong, but the fundamental mindset change that has been taking place for some time is if you are in a situation where there is any chance of creating civilian casualties, or you don't know whether you will create civilian casualties, if you can withdraw from that situation without firing then you must do so."
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