Carey was four not out when he was struck on the grille of his helmet by Jofra Archer‚ a blow that lacerated the Australian’s chin.
After receiving treatment he batted on until the 28th over‚ when he was caught at deep midwicket off Rashid for 46.
Smith marshalled stands of 39 with Glenn Maxwell and 51 with Mitchell Starc before he was run out for 85 in the 48th trying to take a bye.
Woakes took 3/20 off eight overs with Rashid claiming 3/54 from his full quota.
England’s triumph was all but sealed in an opening stand of 174 by Roy and Jonny Bairstow — who have the highest average of all first-wicket pairs in one-day international history.
Bairstow‚ who needed treatment after collapsing with an apparent leg injury in the 12th over‚ recovered to score 34 before being trapped in from by Starc in the 18th.
That was Starc’s 27th wicket of the tournament‚ which broke the World Cup record set by another Australian‚ Glenn McGrath‚ in 2007.
A dozen balls later Roy was given out caught behind by Kumar Dharmasena after sparring at a legside delivery from Pat Cummins.
Replays showed the ball had hit neither bat nor glove‚ but Roy couldn’t be granted a reprieve because Bairstow had used England’s review after he was given out.
Roy stood his ground and remonstrated with Dharmasena and his colleague‚ Marais Erasmus‚ who had to convince the English to leave the crease.