Kyle Walker, making his Villa debut against his first club, scored his first career goal before Marc Albrighton's volley make it 2-0 to the visitors at half-time.
The second half was a different affair however and Jamie Ward brought the Blades back into it from the spot and they had several chances after Ashley Young's dismissal before Stiliyan Petrov's late strike wrapped up the result.
Adams said: "We have got to be pleased that we lost the game with our boots on rather than rolled over and let it be a walkover. In the second half we were terrific.
"I told the boys at half-time that Villa will be thinking the game's won and what we needed to was get a bit closer to them and squeeze up and it worked.
"It's natural for defenders to drop off and not get too close because pace can be a problem but the first goal disappointed me. He ran an awful long way and no one has come and confronted him. We just can't giving teams goal starts."