Wayne Rooney had nodded England into the lead with a second-half header but Ukraine were desperately unlucky not to have equalised after Marko Devic's 62nd minute effort was wrongly adjuged not to have crossed the line.
Gerrard, who will now lead his side into a quarter-final against Italy on Sunday, remained unmoved by the controversy however noting that England had been on the receiving end of a similar error when Frank Lampard's strike against Germany at the 2010 World Cup was disallowed.
"To be successful in these tournaments, because of the standard of teams involved, you need that bit of luck going with you," Gerrard said.
"Two years ago we didn't get that luck with Frank Lampard's goal, a big turning point in that game against Germany, and we ended up packing our bags and going home. Today the luck turned.
"But it just shows that, if you keep working hard and keep fighting, you earn that luck. We deserved it today."
Gerrard also said England deserved credit from emerging on top of a Group D which also included highly rated France and Sweden.
"This was a very difficult group, that was clear when it was drawn out of the hat," Gerrard said. "Not many people have had belief and confidence in the team, but I always kept that belief and confidence because I have very good players next to me in in that dressing room.
"In previous tournaments we've under-performed and not delivered for our country, but in this one so far so good.
"We're working very hard for each other and we showed signs that we are a good team and that we could go on and do well here."
Gerrard, 32, had earlier delivered a man-of-the-match winning performance to steer England into the last eight.
The Liverpool captain led from the front after a rocky first half, creating the opening and the cross in the 48th minute which led to Rooney's goal.
Gerrard admitted England had come out for the second half determined to make an impact after being outplayed in the opening 45 minutes.
"We needed more from each other. We were trying to rally each other. Tonight was an away match against a good passing team, playing in their Euro final in front of their own fans," Gerrard said.
"It was a tough match. When it's not going your way, you have to roll your sleeves up and keep going. We did that in the second half."
Meanwhile England manager Roy Hodgson praised the contribution of his captain.
"You've got to single Steven out because he's just been voted man of the match," Hodgson said. "He's been our man of the match in all three games, playing a captain's innings throughout."