Hodgson used a joke about NASA sending a monkey and a human up to space in a rocket to illustrate the need for England players to get the ball to Tottenham winger Andros Townsend as quickly as possible during Tuesday's 2-0 win at Wembley.
Townsend has called for an end to the controversy, insisting he took Hodgson's comments as a "compliment" and expressing his regret that the subsequent focus on the incident had taken the attention off England's World Cup qualification.
Former England striker Shearer also believes the matter has been blown out of proportion, and told Football Focus: "There's always something with England that wants to sabotage our success.
"The pressure on the players and in particular the manager in the game like this - and at half-time - is absolutely huge.
"But you can't use a word like Roy Hodgson used in a dressing room like that.
"But what I would say is, the time I have spent with Roy, he's an intelligent bloke, he's a tactical genius and he loves his football.
"There was no offence meant, no offence taken. He got the backing - and rightly so - of the FA, and he got the backing - and rightly so - of all the players."
Asked if he felt the story could ever have reached a point where Hodgson's position was called into question, Shearer said: "I don't think it should ever have been under threat, no."
Speculation over who leaked the story to the press has also been rife, but Shearer believes it is likely to have come from a third party rather than one of Hodgson's squad members.
He said: "That's the most worrying thing about it, that whether it's come directly from a player or - what I would think - it's come indirectly. A player has mentioned it to someone else but there's been a breakdown in a player's particular circle (of) who he can trust. That's more worrying."
Shearer was speaking after Townsend had reiterated his support for the England manager.
"It has all been said now, it has been in the news the last few days and I think everyone should focus on us qualifying on Brazil and not focusing on negative, silly news," Townsend told BBC Sport.
"The manager told the player to give the ball to me, so that is a compliment in a way."
The leaking of the joke has sparked a row culminating in a letter of complaint from a pressure group being sent to the FA demanding that Hodgson attend a 'race appreciation' training course.
Peter Herbert, the head of the Society of Black Lawyers and who runs the new Race For Sport organisation, has sent a four-page letter of complaint to the FA, saying it was wrong to declare the matter closed.
The letter from Herbert to FA chairman Greg Dyke states: "To announce that the matter is 'closed' without any action being taken against the England manager is unacceptable and wholly inconsistent with your policies on equality and diversity.
"We are using the appropriate complaints procedure to urge the FA to provide mandatory 'race appreciation' training and 'cultural capital and cultural intelligence' training to Roy Hodgson and all football managers in the UK."