The visitors maintained their perfect start to their Group E campaign thanks to Wayne Rooney's free-kick but looked for much of the game to be struggling to break down their stubborn hosts, who had captain Ragnar Klavan sent off early in the second half for two bookable offences.
Hodgson told Sky Sports News HQ afterwards: "I thought we played some really good football in the first half. I thought the movement was good and there were some very good runs. The chances and half-chances were piling up and when you don't take them, you know it could be a frustrating evening.
"In the second half, there was an element of that. When we scored, we stopped doing a few of the things we'd been doing.
"We knew it wouldn't be easy to win here. We know people will look at the size of the country and say that England should beat them but you've got to play your football.
"There were some very good moments out there - especially in the early part of the game. Strangely that was the part of the game where we couldn't score."
On the performance of Rooney, who missed several good opportunities either side of his match-winning set-piece, Hodgson added: "He'll look back at the game because he studies everything. He studies his own performance and the team's performance, I'm sure he'll say to himself that he could have done better in certain situations.
"Once again, though, we won the game 1-0 and who scored the goal? Our captain, Wayne Rooney. He's once again shown his mettle in that respect.
"I'm just happy that in a week that could have been a banana skin for us, we've got the six points even though we had to work hard for them. We shot ourselves in the foot a little bit by not taking the chances we created."