Wales’ 3-0 win over Russia in Toulouse meant the Three Lions finish second in Group B.
Slovakia must now wait to see if they qualify as one of four best third-place best teams after their goalkeeper Matus Kozacik pulled off a string of saves -- some more luck than judgement.
England will now play their last 16 game in Nice next Monday against the side who finishes second in Group F, most likely against Hungary or Portugal.
This was another frustrating night for England fans in Saint-Etienne. It was at the Stade Geoffroy Guichard where David Beckham was sent off as England lost to Argentina on penalties in the last 16 of the 1998 World Cup.
The England boss boldly made six changes to team, starting strikers Jamie Vardy and Daniel Sturridge, who both scored off the bench in the 2-1 win over Wales.
Hodgson left skipper Wayne Rooney among the replacements with centre-back Gary Cahill taking over the captain’s armband.
Vardy worked hard throughout, but squandered two clear chances of England’s nine first-half attempts on goal to Slovakia’s single effort.
Only England fans could be heard as the second half wore on, nosily urging the Three Lions on in search of the crucial goal which never came.