"We have all got to stick together because there is no-one bigger than the club," stated Cole after notching his first league goal since joining Liverpool from Chelsea last summer.
"I am really happy for everyone at the club; it has been a tough week for everyone - particularly the manager - but we've got the win.
Hodgson 'proud' of Liverpool players
"It is not just the newspapers, none of us like losing and getting criticised but half the battle of modern day football is dealing with pressure and kicking on.
"We are pleased for the fans, the owners and the manager but we have all got to stick together and kick on from here, it can't be a false dawn."
The Liverpool fans turned on Hodgson during and after the Wolves loss on Wednesday, and he was forced to apologise for any offence caused by comments he made about their support for him after that match.
Reports suggest Anfield owners New England Sports Ventures (NESV), which completed its takeover in October 2010, have already started their search for a new manager.
European scout Tor-Kristian Karlsen said on his
feed that Borussia Dortmund manager Jurgen Klopp had been approached about the Liverpool job and that former Hoffenheim coach Ralf Rangnick was also a candidate.
But the Bolton result, which moved Liverpool up to ninth in the Premier League table having played at least one game fewer than most of the teams above them, may have given Hodgson some breathing space.
Hodgson cut a beleaguered figure when Bolton took the lead through Kevin Davies' close-range header shortly before half-time.
But Liverpool equalised just after the re-start through a superb Fernando Torres volley and then pressed hard for a winner.
Bolton did their best to frustrate the hosts, only for substitute Cole to tap home and spark wild celebrations.
Cole appeared to be in an offside position but referee Kevin Friend waved away Bolton's protests as Anfield rejoiced.
"For 90 minutes I thought we were clearly the better team," stated Hodgson. "We had to play well to give that performance and I am naive enough to believe that if you give the performances then often you do win.
"I am extra satisfied we got the winning goal but even at 1-1 I wouldn't have been in any way dejected because I think today we put Wednesday's game well and truly behind us.
"Even if you win it to a controversial goal I don't think we should be in any way apologising for it and anyone who begrudges us that victory after watching the game is a very churlish person."
'Cruel' loss frustrates Coyle
Hodgson revealed he spoke to Liverpool owners John Henry and Tom Werner of NESV following the Wolves match and that they were understanding of his position.
Asked whether he felt he had their support, the 63-year-old replied: "I guess so. They wished me good luck and sympathised with the frustration I must have been feeling.
"It's not about me, it's about Liverpool Football Club and the players. My job is to try and produce good and winning performances from the team.
"Speculation is rife all over, I don't think anybody is immune to it these days. The higher the level you work at, the more speculation there will be.
"Had we won on Wednesday you would now be asking me about the Champions League but because we didn't win on Wednesday I am asked about the negativity which, to be honest, is being created here in this room.
"I know when players are behind their manager and their club and when they're not. Anyone who seriously suggests they are not behind me after this is wrong."
Bolton manager Owen Coyle, whose side lost to a late Maxi Rodriguez goal in the reverse fixture at the Reebok Stadium earlier this season, felt Cole's winner was offside but allowed Hodgson his slice of fortune.
"It's cruel because it happened in the home game," said the 44-year-old. "Maybe that is the little bit of luck Roy needed, who knows, but it certainly went their way today.
"I've not had a chance to look back [at the replay] but I know the players are adamant Cole is offside - he is on the paint as he taps it in."
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