Deila's side host the team from his native Norway in a Europa League Group A match.
Celtic have two points from their opening three games, with Molde leading the section on seven having beaten the Scottish champions last month.
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"It's an enormous game, a must-win match," Sutton, 42, told BBC Scotland.
"To have any chance of reaching the knockout stage they have to beat Molde and then Ajax at home in the next game."
Sutton was asked what it would mean for Deila's future should Celtic not get their first win in the group.
"There would be questions asked if Celtic were outplayed and out-thought by Molde, but let's cross that bridge and wait for the game on Thursday," replied the Englishman.
Former Norwich City, Blackburn Rovers and Chelsea forward Sutton, who scored more than 60 goals in his six years at Celtic, expects Molde to travel to Glasgow with "no fear".
"Most people thought at the outset that Molde were the weakest team in the group," he said.
"I thought Fenerbahce, with all the money they've got, would be the strongest, but it hasn't proven to be that way.
"Molde have turned out to be the surprise package. They've got a game plan and an attacking threat.
"They sat in against Fenerbahce then hit them on the counter attack. Against Celtic, Celtic played well for 10 minutes then that was it. Molde will come to Glasgow with no fear.
"After beating Celtic, they've won two games domestically and they'll fancy it.
"But, from the Celtic point of view, the players have nothing to lose and everything to gain. Celtic should just go for it. They have to win.
"It's a game in which Celtic and Ronny Deila have to make a statement after a bitterly disappointing performance in Norway. Now they have the chance to put it right.
"To do that, first and foremost they have to keep a clean sheet, because there's goals in the team, with [Leigh] Griffiths, and now that [James] Forrest's back and [Kris] Commons is in there. [Jozo] Simunovic at the back should help too."
Celtic's group campaign after they lost to Malmo in the Champions League play-off started with a 2-2 draw against Ajax - a match they finished with 10 men.
They led 2-0 against Fenerbahce in the second round of fixtures, but defensive errors allowed the Turkish side to fight back and claim a draw.
In Molde, defensive frailty was also apparent as Celtic went down 3-1,
"There's been some catastrophic individual errors throughout the European campaign and I understand that Ronnie has to back his players, but some of their individual decision-making has been poor and ultimately it's cost Celtic," Sutton said.
"They can't keep shipping goals - that's been the problem."
Sutton, who played in Celtic's 2003 Uefa Cup final defeat by Porto, feels it's understandable for Deila and Celtic to be judged on their European performances.
"It happened in my time under Martin O'Neill - and to Gordon Strachan, to Neil Lennon," he recalled.
"Europe is an important part of this club and the fans have high expectations.
"It's not unreasonable for fans to expect Celtic to compete in a Europa League group, albeit a tough one.
"The bottom line is that Celtic responded extremely well to that defeat against Molde, beating Dundee United, winning a tricky League Cup tie against Hearts and beating Aberdeen well, but I think fans separate European from domestic football.
"That's just the way it is and this Celtic team need a big performance and a result on Thursday.''

25 November 2015 Last updated at 07:49 GMT
They discovered that mucus and saliva moves in different ways once it leaves your nose.
It moves in sheets, bursts, bags and strings before it becomes droplets.
It's hoped the research will help experts learn more about how sneezing spreads germs.
Take a look at this.
Former striker Chris Sutton believes questions will most likely be asked over Ronny Deila's position if Celtic are "outplayed and out-thought by Molde".
[NEXT_CONCEPT]
Scientists in America have used a high-speed video to find out how snot, or mucus, travels and separates after it leaves our mouths during a sneeze.