Ebbw Vale winger Stephen Hunt scored the winning try in the last second of a pulsating encounter as England buried the disappointments of of Friday to overturn the odds and reach the last four.
England will tackle Fiji later on Saturday after the world champions ended Scotland's run in the cup competition with a 28-5 victory.
Wales, playing in front of the new national head coach Warren Gatland, booked a clash with Australia in the semi-finals of the third-tier bowl competition with a 17-12 victory over Tunisia.
England on Friday scraped into the quarter-finals on points difference, after beating Tunisia, drawing with Canada and losing to South Africa.
They made a blistering start on Saturday with Bristol winger Anthony Elliott scoring from halfway inside 30 seconds.
Samoa then scored twice in quick succession but England hit back just before the interval when Isoa Damudamu sent Harlequins' Tom Guest over to secure a 14-12 lead.
Elliott extended that advantage when he took a quick tap penalty and scorched in for his second try but Samoa drew level at 19-19 with a converted try under the posts.
With the last play of the game, Damudamu slipped a pass out of the tackle for Hunt to score and seal England the win.
Coach Ben Ryan said: "There were times we let ourselves down a bit with our performances yesterday and we were a bit embarrassed. This morning I talked to the players about standards and to show they are worthy of the white shirt.
"I told them that if they don't make it at England Sevens, they won't make it for the Saxons or England seniors.
"I wanted them to front up and they stuck at it and played some really good football against the Hong Kong champions.
"Now we are in the semi-finals and we are excited about that but our goal is to reach the final."