Scotland fans gave their heroes a standing ovation when the final whistle sounded, but cautious Paterson, who snapped up a 31-point haul, urged them not to get carried away with the result.
He said: "Until the last 20 minutes, we were not happy with our performance. But in the end we put a shine on things.
"The aim was to sharpen ourselves up in attack and we definitely created more space. We didn't manage to take all our chances - however we did score some excellent tries."
Paterson, who totted up two touchdowns, all nine conversions and a penalty, added: "One worry is that we missed too many first-up tackles, something we won't get away with against the Springboks.
"We do believe we can go to South Africa and win at least one of the two Tests. We went close the last time we were there and we can complete the job if we focus on the good things from this match and the Six Nations Championship."
Hadden was delighted with his side's display but it looks probable that the win will prove costly.
The biggest worry is back-row Ally Hogg, who was taken off in distress after wrenching a hamstring in the closing minutes of the match.
With Lions breakaway Simon Taylor already sidelined with knee ligament damage, Hadden now faces the prospect of taking on the Springboks in Durban a week on Saturday without his two top loose forwards.
Hadden said: "We won't know the full extent of Ally's injury until he has been more thoroughly examined.
"There are other bumps and bruises and that is a tribute to the Barbarians who gave their all and provided us with exactly the type of physical challenge we needed before the trip."
Hadden confirmed that among the other victims were centre Marcus di Rollo (knee strain), lock Scott Murray (bruised ribs) and full-back Hugo Southwell (dead leg).
But it is the plight of Hogg that will cause the most concern. He collapsed in a heap with no rival player anywhere near him during the move that led to the final Scotland touchdown from Paterson.