Wasps stunned favourites Leicester 25-9 before a world-record club attendance of 81,076 at Twickenham to seize their second European crown in four years.
The Tigers were heavily fancied to complete the final leg of an unprecedented treble but they were outmuscled throughout in a surprisingly one-sided final.
It was a remarkable performance which Dallaglio claimed highlighted his side's 'soul' - although he is concerned it may become impossible to sustain what has become a golden era for Wasps.
He said: "At the moment we don't have the resources or financial clout that Leicester have.
"That's something we've got to sort out. We haven't got the buying power or strength in depth that they have.
"We don't own a stadium or have 17,000 people coming to watch us. But what we do have in our team is soul.
"That's what we've always had and that's why we've stood toe to toe with them in the professional era.
"We need to go away and address the issues off the field and make sure we don't fail because of our lack of ability to compete at that level.
"We need to sort ourselves out and move forward. We need to compete with the likes of Leicester, Harlequins and Northampton who have great facilities."
Eoin Reddan and Raphael Ibanez scored identical first-half tries in a nightmare 20 minutes Leicester will want to forget as quickly as possible.
First Reddan exploited an off-guard Leicester pack to scamper home after taking the ball from close range and then Wasps pounced from the same position for a second time.
On this occasion Ibanez combined with Simon Shaw before diving in at the right corner for a try that had fans shaking their heads in disbelief that the Tigers had been caught out twice.
Wasps fly-half Alex King finished with 15 points despite missing four shots at goal with Wasps not conceding a single point in the second half.
The match kicked off shortly after news emerged that English and French clubs had agreed to take part in next year's Heineken Cup and European Challenge Cup tournaments.
Anglo-French sides announced a boycott of the competitions nearly two months ago following a row over shareholding rights but talks, notably involving Premier Rugby and the Rugby Football Union, have resulted in a breakthrough.
Dallaglio said: "The match was a great advert for the European Cup. I often think finals can be bit of a letdown but I don't think this one was.
"We're obviously all thrilled to bits that the Heineken Cup will go ahead next season - as will be every country that participates.
"We've had a record attendance for the final this year and it's a fantastic competition."
Leicester skipper Martin Corry insists his side could have given no more in their quest to add European honours to their Guinness Premiership and EDF Energy Cup crowns.
He said: "In the second half we gave everything - I saw that in the lads' eyes. As a captain and coach that's all you can ask for.
"It's been a long season but we were fired up and fresh for today - we were just beaten by the better side."