Paul Hayes struck the foot of the post before pulling a fantastic save out of Begovic in the opening five minutes.
Adkins was pleased with United's dominance, saying:"It was a game we controlled for the 90 minutes and we had plenty of chances"
"Some of our football was outstanding. We had two spells of pressure in the first half that were exilerating to say the least.
The United boss is hoping his side can manage the expectation and claimed they will "work hard" on keeping up the good form.
He added: "We all want to do well this season and we have given ourselves an opportunity to do that."
The Yeovil Town manager Russell Slade bemoaned two missed chances before the brwak.
He said: "Scunthorpe is a difficult place to come and not many teams will come here and do well.
"But we had a couple of chances before the interval and, if we'd taken one of those, then it may have been a different story."