"That is really our only chance of going further, I believe," he told a news conference on Tuesday. "We haven't got a lot left to lose anymore. Our joker is already gone. If you want to progress it's about winning matches."
"It's about stepping up now and hoping that we're more effective in front of goal. Then it can go well."
Kallstrom was addressing a muted news conference at the Swedish team's base on the outskirts of Kiev, the day after they had allowed Ukraine to come from a goal down to win 2-1.
The defeat left Sweden at the foot of the table behind France and England, who drew 1-1 in the other Group D game.
"The players must grieve today and I will too, but tomorrow we start with new thoughts and concentrate on England," coach Erik Hamren said.
After a poor first half, the Swedes took the lead shortly after half-time through their captain Zlatan Ibrahimovic, but two headed goals from Andriy Shevchenko gave the hosts victory.
Johan Elmander and Olof Mellberg missed great chances to equalise late on.
"I think we are worthy of at least a draw, but we leave there without a point. It's a tough day and it should be," said Hamren.
"It's small situations that decide if we lose the game yesterday or not," he added.
There were few signs of the free-flowing football Sweden had played in winning all four of their matches this year.
"The man in possession of the ball had too few alternatives. That's what I brought up at half-time," said Hamren. "The man in possession saw mostly backs, there were very few who wanted the ball."
Hamren said at the post-match news conference that his players had "lacked courage and been a little cowardly".
"We will obviously have to show more [courage] against England," midfielder Rasmus Elm said. "Yesterday there was maybe a little too much uncertainty.
"It doesn't sound like Erik to use the word 'cowards' but I think we need to step up and show that we want it. If we do that, we can beat England."
