Everton double sink Crystal palace
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| Dominic Calvert-Lewin has scored 12 goals in all competitions since the start of last season, more than any other Everton player |
Tactical changes by Toffees boss Marco Silva decided the game, with Dominic Calvert-Lewin heading in fellow substitute Ademola Lookman's cross.
Moments later a third substitute, Cenk Tosun, hit the winner at Goodison Park.
"Had we won or drawn no-one would have begrudged us that," Hodgson said.
Earlier in the second half, Everton keeper Jordan Pickford saved Luka Milivojevic's penalty with his feet.
Everton's third victory in a row lifted them to eighth in the table, while Palace stay 15th, two points above the relegation zone with only two wins this season.
Hodgson's team, who have only scored five goals this term, face a tough run of fixtures over the next month, starting with next Sunday's home game against fifth-placed Arsenal.
Palace then go to Chelsea the following weekend and host Tottenham on 10 November, before returning after the international break with a trip to Manchester United.
"Football does not give you what you deserve, it is a question of results. All you can do is hope the team played well enough to get a result," Hodgson said.
"Every game in the Premier League is a tough assignment. Obviously the top teams have very good quality players and have more resources than some of teams we play against.
"But the bottom line is have to play all the teams in the league. Everton have qualities and we were prepared to try and nullify them. We will do same against Arsenal."
Silva strikes gold with substitutes
For most of the afternoon a rather drab encounter looked destined to finish goalless as both sides struggled to create clear chances.
What swung the game in Everton's favour was Silva's decision to make a bold attacking double change, bringing on forwards Calvert-Lewin and Lookman with eight minutes left.
The home side effectively switched to a 4-2-4 formation, having earlier introduced striker Tosun for attacking midfielder Bernard.
Lookman clipped in a precise, curling ball from the left which was met by a Calvert-Lewin header from close range.
As Palace belatedly tried to respond, Tosun latched on to a long punt clear by defender Michael Keane to fire through Palace keeper Wayne Hennessey's legs one minute 44 seconds later.
"It came down to which manager was brave enough to make the big changes - and it was Marco Silva," former Scotland winger Pat Nevin said on BBC Radio 5 live.
"Luck favoured the brave. I don't think you will see too many better substitutions this season."
This was the first time since August 2017 that three substitutes were directly involved in goals in a Premier League game.
"We, the staff, are here to make decisions, it is not just the starting 11," Silva said.
"The three players who came on were aggressive, it is really good and I'm very happy for them."
England international Pickford, 24, attracted plaudits during the international break after his distribution helped set up two goals in the surprise 3-2 Nations League win over Spain on Monday.
Again he produced a moment of quality with his feet, albeit in a rather different manner after Seamus Coleman tripped Wilfried Zaha to concede the penalty.
Diving to his left, Pickford stuck out his right boot to stop Milivojevic's spot-kick down the middle and prevent Palace from taking the lead with only their second shot on target.
It was only Milivojevic's second penalty miss in his past 11 attempts.
"It was a good save from the goalkeeper," Crystal Palace manager Roy Hodgson said. "If Luka had scored it might have been a different story."
Wickham returns after two-year absence
One positive for Palace was the return of striker Connor Wickham after a two-year absence because of a serious knee injury and subsequent problems.
The 25-year-old made his first appearance since November 2016 when he came him on seconds after Tosun's 89th-minute goal, but had little time to make an impact.
"He has worked so very hard to get into that position where he is deemed fit enough to get on the bench," Hodgson said.
"We would not have risked him for a long period of the game. I'm sure he will be pleased to have got on the field for Palace in a Premier League game and hopefully it will give him hope and heart.
"He has still got a lot of hard work to do on the training field."
Man of the match - Jordan Pickford (Everton)
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