Gary Hooper gave the Iron the perfect start with a goal inside the opening 80 seconds before Grant McCann got his reward for a faultless display with a second goal 13 minutes from the end.
The goals brought about the Iron's maiden home league win over Wednesday, and more importantly moved them above their opponents in the Championship table and six points clear of the relegation zone.
Adkins said: "We knew that this game would be the hardest of the season and a defining game in the season. We got off to a great start and after that we had to defend manfully. We made poor decisions, simply because of the effects of the weekend.
"The lads had to show fantastic character. It was against a team in form, a team that gave us a good hiding earlier in the season. To win the game 2-0 just shows how far we've come.
"We have a group of players that have come a long way and we've a lot of ability in this team. We knew it would be a tough game as it proved to be. Sheffield Wednesday dominated the game but we defended very well.
"When you've had a big game, it's the game after that is always difficult because of the mental energy you exert. We've just beaten Sheffield Wednesday at home, we're Scunthorpe United and that's fantastic."
Wednesday manager Alan Irvine was disappointed to see his perfect start to life at Hillsborough come to an end and was also left to rue his side's inability to trouble Scunthorpe goalkeeper Joe Murphy.
Irvine had won his first three games in charge of the Owls but was helpless as his men laboured in front of goal, despite plenty of possession and pressure.
They created just one chance of note during a disappointing first half and did little better after the interval, with Michael Gray fluffing his lines when clean through in what turned out to be their golden opportunity.
"I think we deserved something out of the game. It was a horrendous start and I'm disappointed about that," said Irvine.
"I'm not going to be too critical, we will hopefully learn. We were on the back foot for the first 20 minutes or so and I thought we got a grip of the game from then on.
"We got in the final third a lot and you have to give credit to Scunthorpe, they defended their box very well. Maybe we needed a bit more quality to open them up. We huffed and puffed a lot but we didn't look like we were going to score a lot of goals.
"The lads realise where we are. We're obviously in a relegation battle and are aware of that. We need to keep that in mind until we get ourselves safe, if we're able to do that."