Adam was a key figure for the Staffordshire outfit in their 2-0 Barclays Premier League victory over Sunderland on Saturday.
The midfielder scored the opening goal on the half hour and was involved in the build-up to Steven Nzonzi's strike that wrapped up the three points for the hosts in the 81st minute.
Adam had been restored to the first XI after starting on the bench for the last three league games.
In Stoke's previous outing he had come off it and kept his cool to convert a stoppage-time penalty that earned his side a 3-3 draw at Swansea.
And Arnautovic is in no doubt Adam should now be a regular fixture in the starting line-up.
Asked about the Scotland international's goal against Sunderland, Arnautovic said: "It was important for him.
"In the last few games he has not played from the start and now he has showed that he has to play."
Adam was also involved in the most controversial moment in the game when he was challenged by Wes Brown in the 36th minute.
Referee Kevin Friend deemed the sliding tackle from Brown that won the Sunderland defender the ball - and replays of which suggested any contact between the players had been minimal - to be worthy of a straight red card, a decision that left Black Cats boss Gus Poyet fuming.
Arnautovic claimed he did not see the incident, but said of the match overall: "I think we did well but we were also lucky because they got a red card.
"They had one man less than us and then they tried to push because they wanted to score a goal, so we went on the counter-attack and Steven's was a nice goal."
The Austria international added: "I can only say I'm happy to have these three points and win this game.
"We have to keep playing like this so we can keep going up the table. We need the points."
Having recorded their first win in nine league matches, Hughes' side, who had been 17th at the start of the day and separated from the relegation zone by goal difference alone, moved up to 14th on 13 points from 12 games.
While they are three points clear of the drop zone, Sunderland are three points adrift of safety and have slipped a place to bottom spot.
Poyet, in charge of them now for five league games since succeeding the sacked Paolo Di Canio, had plenty to say about the red card incident after the contest.
The Uruguayan manager called for an apology from referees' chief Mike Riley - West Brom boss Steve Clarke recently said he had received one - and the Black Cats are expected to appeal.
But Poyet would not say that the decision had definitely cost his side the game, and added: "We need to change things.
"We need to improve and keep getting better, make sure we don't concede so that we stay in the game and can win it, keep believing in the game we are playing.
"I think we are better, but the problem is that we need to be a little bit better again to win games away from home. We have not done it once this season."
Sunderland have won both their home league games under Poyet and lost all three away matches.