Walcott levelled in the 64th minute with a fierce shot and he delivered a fine cross 12 minutes from time for striker Danny Welbeck to tuck the ball in with a clever backheel and clinch England's first competitive victory over the Swedes.
England forward Andy Carroll had opened the scoring with a superb header midway through the first half but Sweden struck back after halftime thanks to a Glen Johnson own goal and Olof Mellberg's header.
England are joint top of Group D on four points with France who beat Ukraine 2-0 earlier on Friday.
Ukraine, who play England in the last round of matches on Tuesday, are third on three points and Sweden have none.
"It was my first appearance in such a high-profile arena," Walcott said. "I've been waiting a while. I hope there are more chances for me."
Coach Roy Hodgson had brought in Carroll at the expense of Alex Oxlade-Chamberlain to exploit Sweden's weakness in the air and the choice paid off spectacularly midway through the first half.
Steven Gerrard whisked in a perfect cross for Carroll to head powerfully home from near the penalty spot and give England a deserved advantage.
It spurred Sweden into action but England initially remained composed.
That calmness vanished, however, early in the second half when the unmarked Mellberg bundled the ball in after the English defence had failed to clear an Ibrahimovic attempt in the 49th minute.
Johnson tried to clear on his line but only could help the ball into the net.
England were on the ropes and Mellberg delivered another body blow 10 minutes later when found himself unmarked to head in Sebastian Larsson's free kick from close range.
Defenders Joleon Lescott and John Terry were nowhere to be seen and Joe Hart was left stranded on his line.
Sweden's Andreas Isaksson looked much more alert when he denied Terry with a quick reaction save and Walcott replaced the ineffective James Milner in the 61St minute to make an immediate impact.
Sweden failed to clear a corner and the winger's fierce shot swerved high into the net past a static Isaksson.
Walcott then broke clear down the right flank to set up Welbeck who neatly diverted the ball into the far corner with his back to goal and England hung on without undue alarm to take the three points.
