Teenager Owen Farrell, son of Sarries head coach and former Great Britain rugby league captain Andy Farrell, kicked them to a 22-18 victory over Leicester at Twickenham.
"The difficult bit is going on from here," said Saracens' multi-millionaire chairman Wray, whose financial backing and fierce loyalty has been at the club's core since 1995.
"I am a huge believer in life of that phrase 'show me a complacent man and I will show you a failure'. You can't be complacent - you have got to go forward.
"Over the years, there have been many sleepless nights and many disasters, but if you don't fail you don't know what success is. You have to fail to enjoy success."
Sarries' nerve-shredding win on Saturday denied the Tigers an eighth Premiership title and consigned them to a rare season without silverware.
It also gave Saracens a first major trophy since world superstars Francois Pienaar, Michael Lynagh and Philippe Sella inspired an emphatic Tetley's Bitter Cup final triumph over Wasps 13 years ago.
And it meant Saracens joining an exclusive club, becoming just the fifth Premiership winners after Leicester, Wasps, Sale and Newcastle.
But Wray does not view the Hertfordshire-based club's finest triumph since being formed in 1876 as a case of mission accomplished.
He said: "What we've got to do now is go on from here because the moment you pause for breath and think you are bloody marvellous, you are dead.
"We will need to work harder and see how we can get getter. We do a lot of things well, but we need to everything better.
"There are no guarantees in life, but the thing we must not do is rest on laurels."
It has been a rollercoaster ride for Wray during the past decade and a half.
Beaten by Leicester in last year's final and play-off qualifiers in 2007, Saracens' Premiership history also shows three 10th-place finishes and a flirtation with relegation.
"If I had known what it would cost in terms of money and also emotionally, then of course I wouldn't have done it. I would have just rolled up as a fan and let someone else do it," he said.
"But having done it, I don't regret a thing. It is a huge privilege to be able to have these opportunities and to try to create something special.
"Over the course of a journey we all get disheartened. It's just about, can you keep going? You pick yourself up and you get on with it.
"I think the essence of life is taking a risk, and I like taking risks. I don't like getting it wrong, but I want to be in the arena.
"I've always described myself as an overnight success - it has only taken 16 years!"
Teenage fly-half Farrell underpinned Saracens' success with 17 points, included the conversion of wing James Short's first-half try, for a 100% goalkicking return.
But it was the sustained brilliance of hooker Schalk Brits, who created Short's touchdown, and an heroic defensive rearguard when Leicester went through 30-plus phases in injury time that defined Saracens' momentous day.
"In that last 10 minutes, you could see what this team is about - it was sheer courage," added Wray.
"If you don't want to give in, you don't, and seeing the guys give everything like that was fantastic.
"They are an incredible bunch of guys. Under that pressure I would have buckled, but they didn't, and we are very lucky to have such an outstanding group of people."
Brits said: "After last year's final there were a few things we wanted to change. We are one of the best defensive sides in the Premiership.
"I had no doubt the boys would keep them out - we had been in the same position a number of times this season. I just know the guy next to me wants to put his body on the line as much as me.
"We have worked very hard for two years to reach this title. We have created a bond in this team. Some people see it as work, but we see it as creating new memories."
Saracens' workaholic Namibian flanker Jacques Burger underlined the desire to build on Premiership success next season, both at home and in Europe.
"You don't want to be a flash in the pan," he said. "We want to win the Premiership again and do well in the Heineken Cup.
"You don't want to be amazing one year, then not play any good rugby the next season. We are going to be humble and work hard.
"The belief has come through this season, and when you are in tough situations it's the belief that gets you through."
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