Niall McGinn scored the only goal of the game with an 18th-minute solo effort, and Smith felt it summed up his side's shortcomings, as they languish adrift at the foot of the table.
"It's a hard one to take because I think we dominated the second half and created chances in the first half. I can't fault the boys for effort, we just need a wee bit more quality in the final third," he said. "It's just the finer details we're not doing well enough at this moment in time."
For the match-winner, McGinn received the ball on the halfway line and shuffled past two men before unleashing a low, 20-yard drive which bounced just in front of Rab Douglas, and nestled in the back of the net.
Smith said: "We've spoken about it [the goal] in the dressing room. These are the small margins that win or lose games. We switched off for a very short period of time, lose a goal, and then the chances that come along should be taken."
Aberdeen boss Craig Brown admitted his side perhaps had luck on their side at Pittodrie as they took all three points.
The former Scotland boss felt that the result possibly made up for matches the Dons should have won, but could only draw.
He claimed his side could have had a first-half penalty which might have made the game more comfortable, and said: "The second half from our point of view was poor; the first half was good.
"I thought we played well, and there was a clear handling offence on Scott Vernon's shot which would hopefully have given us a more comfortable lead.
"The performance wasn't good in the second half, and it kind of makes up for a fortnight ago against Hibs. We dominated that game and only got one point whereas in the second half Dundee dominated and we got three points, so it perhaps evens it out a bit."