The result edges England a point clear in Group H.
It means victories at Wembley against Montenegro and Poland will sweep the Three Lions into next summer's World Cup in Brazil.
The margins are tight though, with the loss of any points likely to condemn England to the play-offs, or possibly even worse.
And given they have drawn all four matches against meaningful opposition in this campaign, it is no wonder some are casting a critical eye over Hodgson's team.
Chief amongst them is Lineker, who in a series of postings on Twitter during the match, bemoaned England's inability to keep the ball and questioned why Manchester United midfielder Michael Carrick had not been introduced.
"We're seeing a lot of one touch football. England have a touch. Ukraine have a touch. England have a touch...," said Lineker.
"When you see England pass it like they've never been introduced to each other, it does make you wonder why Carrick doesn't feature.
"We're on the precipice here. KEEP THE BALL."
In addition, in a Tweet he subsequently deleted, Lineker said: "Awful. What happens to some of these players when they pull on an England shirt?"
Finally, Lineker said, with more than a hint of sarcasm: "Good result for England....Brazil, Spain, Germany, Argentina et al, you're in big trouble!"
When informed of the criticism, Hodgson was unimpressed.
"I cannot comment on what people think at home, I don't think that is a fair comment on our performance," he said.
"If the expectations upon us are so great we come to places like Ukraine and expect roll teams over, it will be difficult for us to ever get a good result.
"I will stick to my ignorance. I think it is a good result. I am surprised that has not been the reaction back at home.
"Who are we talking about? There are 60million people back there. I find it a difficult one to answer.
"Before coming here, the feeling amongst you people was that a draw would have been a good result.
"I am surprised to hear people say - what I thought was a good performance - has become yet again underwhelming."
England's best performances were undoubtedly at the back.
Gary Cahill was outstanding, with central defensive partner Phil Jagielka barely less impressive.
Ashley Cole used all his experience to shut down one flank, whilst even Kyle Walker emerged with credit given he had the task of repelling the threat of Ukraine danger-man Yevgeni Konoplienka.
"I am happy," said Hodgson. "In the first half we played well but although we came under more pressure for a short period in the second half the back four stood very firm.
"Attacking-wise our options were limited as three of our top strikers were not available and we also had (Alex) Oxlade-Chamberlain also unavailable.
"I thought the players who took their place did a remarkable job."
So now it boils down to two games at Wembley next month.
England could go through by winning the first of them. It is unlikely though, with Hodgson accepting the situation is incredibly tight.
"It is still very tight," said Hodgson. "We knew a victory would put us in a good position and a draw would maintain our leadership and make certain going into the last two matches at Wembley we have our fate in our hands.
"I am more than happy we go into these last two games at Wembley top of group and one more point than our rivals.
"It is an unbelievably tight group and that is positive as far as we are concerned."
Not that you would know it by listening to the musings of Hodgson's opposite number Mykhailo Fomenko.
"It is difficult to assess our chances," he said.
"We played a strong team. They tried and we defended. We tried and they defended. Then we ended up with a result."
- Sports & Recreation