The Valiants had recorded victories over both Sheffield clubs in the previous rounds but Adams' men bowed out of the competition after strikes from Paul Hayes and Grant McCann in the third-round tie at Glanford Park.
Adams said: "I think we can be proud of what we have done, with a bit of luck, it might have gone our way.
"We had chances in the first half and we had chances in extra time after they scored. Their keeper played really well in extra time. It just wasn't our night."
Adams is now urging his side to take the momentum from their impressive cup run and translate that into results in League Two, where they currently sit in mid-table.
"It's all important now, we've got to take a great deal of confidence from this run and push on in the league now," he added.