Gary Hooper had put the Iron ahead with a debut goal, but Enoch Showumni levelled for the visitors shortly afterwards and Jermaine Beckford got the winning goal 10 minutes from time.
Scunthorpe had been the better side in the opening half, and although the visitors came back strong after the break Adkins felt his side's performance should have merited a point.
"Credit to Leeds, they've upped the tempo in the last half hour of the game and Josh Lillis has had to pull off some fantastic saves," he said.
"We're disappointed with the first goal we conceded because we didn't expect to concede like that. And the second goal was back off the crossbar and into the path of one of their players to score.
"We're disappointed that, with the effort we have put into the game, we have not taken any points from it.
"We have to make sure that when we do get a lead, we hang on to it. It's important that we know how to see a game out."
The Leeds manager Gary McAllister stressed that the campaign is "a marathon not a sprint", and felt the three points heading back down the M62 were well deserved.
"Coming here is not easy, it's a tough place to come. But we're sending out the right messages," he said.
"In the second half we got a bit of rhythm and by the final whistle, I thought we deserved the victory. We worked hard as a team and got our just rewards."