The 38-year-old only took charge at Parkhead in June, but he has already come under pressure following Celtic's 2-1 aggregate defeat to NK Maribor in the third qualifying round of the Champions League, which demoted the Scots to European club football's second competition.
However, ahead of this evening's group opener against Red Bull Salzburg, Sutton, who was speaking to Sports Mole to promote BT Sport's live coverage of the encounter, has suggested that Deila could win over the doubters if Celtic have a good run in the Europa League.
"It has been a baptism of fire because it's a real tough job to come in to. There are big expectations and there always has been at Celtic. He has come in and talked about his philosophy, which seems to be attacking football. Celtic have always played attacking football, so that matches," said the 41-year-old.
"Football is a results business and Celtic fans have been disappointed, there is no doubt about that, with the Champions League campaign. They feel, quite rightly, that their side faced a team that they should be beating.
"Now he has his own players in, it's time for the team to start performing. He's got the weapons in terms of pace and physical presence, which is crucial, so now people will start to judge him more and take a closer look at his work. This Europa League campaign will absolutely crucial to him and the way that he is perceived at Celtic."
Read Sports Mole's preview of Celtic's clash in Austria with Salzburg here.
BT Sport has live coverage of UEFA Europa League games throughout the group stages and beyond, including Salzburg vs. Celtic this Thursday on BT Sport 1 (not in Republic of Ireland). Visit www.btsport.com to find out more.