Wales’ 3-0 win over Russia in Toulouse meant the Three Lions dropped to second in the group, behind their neighbours.
Slovakia must now wait to see if they qualify as one of four best third-place teams after their goalkeeper Matus Kozacik pulled off a string of saves — some more luck than judgement.
The Slovakia shot-stopper was man-of-the-match and after the match he backed the England to progress past the last 16.
England will now play their last 16 game in Nice next Monday against the side who finishes second in Group F, with Hungary, Portugal, Iceland and Austria all still in contention.
“It’s disappointing that we had all the play and had all chances around their box, I can’t deny that, but we have players who can score goals and will score goals,” insisted Hodgson.
“Yet again, this is a game we should have won, because everything we did on the field put us in a position to win, but we didn’t use those corners and free-kicks to get the win.
“It’s a frustration myself and the players will have to live with.”
This was another night of opporunities missed 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.
Prince William, who turns 34 on Tuesday, was in Saint-Etienne, sat next to FA chairman Greg Dyke who has said Hodgson will remain England manager — provided they reach the semi-finals.
Hodgson’s contract expires after Euro 2016.
The England boss boldly made six changes to the 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.
Slovakia coach Jan Kozak kept faith with the starting side which beat Russia 2-1 last Wednesday.
Dominance not enough
Vardy worked hard throughout, but squandered two clear chances of England’s nine first-half attempts on goal to Slovakia’s single effort.
He fired over early on, then had his shot blocked by Kozacik in a one-on-one after winning a sprint against Slovakia captain Martin Skrtel.
And only a fine tackle from right-back Peter Pekarik denied Sturridge at the near post in the opening 10 minutes.
Lallana was then denied by Kozacik’s reflex save with half an hour gone.
England’s frustration continued after the break.
A mix-up between centre-back Chris Smalling and goalkeeper Joe Hart allowed Slovakia winger Robert Mak a half-chance.
At the other end, only the slightest of deflections off Kozacik’s shoulder kept out Nathaniel Clyne’s shot.
There was a big cheer from travelling England supporters — the loudest of the night — when Wayne Rooney came on for Jack Wilshere with 56 minutes gone.
Dele Alli, on for Lallana, then saw his shot cleared by Skrtel after finding himself unmarked when he arrived in the area at pace.
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.
Rooney was a constant menace after coming on, firing wide as the game entered the final 10 minutes while Harry Kane, on for Sturridge, had a couple of late chances — all to no avail.
For Slovakia, a point was as good as a win and their players celebrated after the final whistle.
“To get a point against England is a huge positive,” said Kozak.
“I am glad to have four points, but we have to see how things progress and if we get through to the last 16, we’ll be really happy.
“We tried to attack, but the quality of the opposition, especially late on, meant we were deep in defence and couldn’t get out of our half.”