Summer signing Francisco Sandaza scored the winner from the penalty spot at the start of the second half.
Boss Craig Levein told Setanta Sports 1: "I'm thrilled. I think the result was everything for us today.
"The way the players dug in when we got in front was pleasing to see because no doubt we need that kind off determination again in future matches."
He added: "When you go to Pittodrie, it's always difficult but, thankfully, we've got the win today.
"It was arguable whether either team deserved to win in all honesty.
"But we got a break, we haven't been getting them up to now, and I'm grateful, we're relieved, and the players are delighted with themselves."