Ukraine takes on Group E winners Sweden in Glasgow in the Round of 16 on Tuesday.
Sweden won two games and drew one in the group phase while Ukraine only managed a win over North Macedonia, after a 3-2 loss to the Netherlands. They were also beaten 1-0 by Austria.
Ukraine were the fourth-ranked of the six group third-placed sides, meaning they scraped into the last 16 where they are underdogs to Sweden who made the 2018 World Cup quarter-finals.
“We achieved the result we wanted,” Shevchenko said. “We have nothing to lose now. Everything else will be a big bonus for us.
“The team played very well the first two games, the third game not everything went as we planned.
“We achieved the result we wanted. Now we have another game to get even more and we’ll do everything we can to achieve a good result.”
Former AC Milan and Chelsea star Shevchenko offered praise to Sweden, noting their star strikers Marcus Berg and Alexander Isak.
“Sweden are a very organised team, and I am full of respect for them,” he said.
“They have a certain style of play, and they don’t deviate from it. The things they do they do very well, especially their passes upfield.
“They have two great strikers, and are very strong physically so rarely lose one-on-one battles.”
Sweden boss Janne Andersson implored his side not to under-estimate Ukraine, despite battling against Austria last time out.
“Ukraine are a good team; they know how to switch from defence to attack,” he said.
“They looked worn out against Austria, but like us they’ve had some rest even more actually.”