Adams was appointed on Sunday night following the decision to end Chris Hughton's 20-month reign after a string of disappointing results left the Canaries 17th in the table and five points clear of the relegation zone.
Hughton had been under pressure at different periods of a season in which a summer spending spree never truly inspired his side to build on an 11th-placed finish last year, but the final straw was a home defeat to fellow strugglers West Brom on Saturday.
Norwich now face a trip to Fulham, the team directly below them in the table, before a final run of four fixtures that sees them play Liverpool, Manchester United, Chelsea and Arsenal.
But McNally wants Adams to pick up enough points in those daunting fixtures to both keep the club in the top flight and seal his position on the bench beyond the end of the campaign.
"We will have to wait until the end of the season but it is all in Neil's hands, if he wants the big job here it is entirely up to him," he said.
Having won just once in their last six games Norwich have slowly slipped into the relegation dogfight and McNally believes the decision to change was not as 'bonkers' as the likes of Gary Lineker have suggested.
Lineker, Hughton's former Tottenham team-mate, Tweeted on Sunday evening: "Chris Hughton has been sacked by Norwich. Now? With 5 games to go? Utterly bonkers!"
But McNally said the decision was taken with the club's supporters in mind.
"We felt we were left with no choice," he said.
"Recent form and performances have been disappointing, We have lost four of the last six games and have lost seven away games consecutively.
"We had to do something to ensure we have the best possible chance of staying in this league. As a football club we are very supportive of our mangers, we give them time to get the job done and do things right.
"Clearly it is a results business and four points from the last six games, particularly against peer-group clubs, hasn't been good enough. We believe that with Neil in charge for the next five games we have got a greater chance of improving our position.
"If you read the reaction over the past 24 hours, the national media have said we are bonkers but a lot of Norwich supporters have been encouraging us to make this decision.
"We are only here for two things, we are here for the football club and for the supporters, we have to do everything we can as often as we can."
Adams himself takes the job having led Norwich to the FA Youth Cup last season and, having played for the Norfolk side for five years before moving into a coaching role, he jumped at the chance to prove himself at senior level.
When asked how long it took him to accept McNally's offer of replacing Hughton, the 48-year-old replied: "Two seconds. It was an absolute no-brainer of course.
"I have been here almost 20 years as a player and a coach and to be offered the opportunity to manager the club is an honour."
Hughton's last game saw the Norwich supporters turn on him at full-time, jeering the result and throwing a number of paper clappers - aimed at improving the atmosphere at Carrow Road - in his direction.
Adams admits witnessing that spectacle was not a pleasant experience but thinks he can do enough to avoid facing a similar demise.
"It is never nice," he said.
"It was unsavoury and it was a disappointing time for everybody and that disappointment has obviously come from the result which is the primary thing we are addressing, I am confident I can get the results to keep us in this division."