England's 4-0 win over Moldova on Friday night put them top of Group H on goal difference after second-place Montenegro could only draw 1-1 at home to Poland.
As they have played one more game, Montenegro do not have a match on Tuesday, meaning England would remain top of the table if they draw in Kiev.
But Walcott thinks holding out for a point in the Olympic Stadium would be the wrong thing to do.
"We always go in to games trying to win them. We don't want to go out there and try to get a draw," the Arsenal winger said. "That's when we could be caught on the break.
"We have a lot of pace in this team. They will have to come at us and I think that could work in our favour."
Kiev holds fond memories for Walcott as it played host to the stunning volley he scored against Sweden at Euro 2012.
He is looking forward to going back to the Ukrainian capital, but is aware that the hosts will be desperate to secure another victory having boosted their goal difference with a 9-0 win over San Marino on Friday.
"It's going to be really nice looking for that goal again and I go there feeling very confident as do the team as well," the 24-year-old said.
"That was a great occasion for myself scoring that night but I need to concentrate on getting the points now on Tuesday.
"They did very well on Friday. They scored nine goals so they will go in to the match on a positive note."
After an unsettled beginning to last season, Walcott has found himself very much central to the plans of both Arsene Wenger and Roy Hodgson this term.
The 24-year-old is not the only Arsenal player to have impressed the England manager, however.
Hodgson spoke of his delight at having a fit-again Jack Wilshere available for the game against Moldova, where he combined well with Steven Gerrard and Frank Lampard in the midfield.
Debate has raged over the last decade about the supposed inability of Gerrard and Lampard to play alongside each other in midfield, but Walcott thinks Wilshere's presence helped the two old heads combine well at Wembley.
"I bet Frank and Stevie feel very old next to Jack," Walcott said with a smile.
"It definitely did work, they communicated well and blended well, the rotation worked well and they all had a part to play in the (Gerrard) goal so there are a lot of positives to take from that.
"We have lots of players in that position but they did very well against Moldova."
Wilshere had to sit out training on Tuesday with a stomach bug and there are still concerns about his fitness following a lengthy ankle injury layoff, but Walcott is convinced the 21-year-old will be fit for Tuesday.
Walcott added: "Jack is a strong lad, he always wants to play football and I am sure he will be absolutely fine. He will recover."