Grant has failed to win any of his opening eight league matches in charge, with four draws and four defeats leaving the Pars rooted to the bottom of the cinch Championship table.
The former Alloa Athletic boss felt the ire of the travelling support after the loss in Dumfries with angry scenes greeting the 56-year-old as he made his way across the pitch to the away dressing room at full-time.
Fans have called on the club's new German owners to swing the axe, however, in a statement released on Tuesday, the East End Park board of directors backed Grant to turn things around, and criticised the behaviour of supporters.
It read: "The Board would like to state that the behaviour of a handful of supporters at the end of Saturday’s match was completely unacceptable, and as has been highlighted we can accept everyone is frustrated and disappointed, or indeed angered by not winning matches, but there can be no excuse for that type of behaviour.
"In addition, the original members of the DAFC Board are made up of supporters themselves who have acted on a voluntary basis for the last eight years, but in recent weeks the level of abuse directed to them has also become totally unacceptable.
"Now is not the time to say any more on that subject, as our energy has to be focussed on winning football matches. However, the events of the last few weeks have certainly not made for a positive portrayal of our club to our new investors and potential new sponsors."
Dunfermline sporting director, Thomas Meggle, also gave his personal backing to Grant, insisiting that "issues outwith the manager's control" had contributed to the early season struggles.
"It is important to keep a sense of perspective at all times," he said. "Covid interruptions, bad luck with injuries and issues out with the Manager’s control are not excuses, they are valid reasons.
"Unfortunately, we lost Saturday’s game against Queen of the South when we could easily have won the match if a legitimate goal had not been disallowed. I can understand the frustration of the fans, as I am disappointed with the way the season is going as well.
"However, I see how much the team is already fighting for the club and the team, and Peter and the coaching team are working tirelessly to turn things around. I believe that we can only overcome this situation by all of us sticking together."
"It is always easy to find someone to blame. In football, it`s usually the coach. But after my analysis, I believe we need time for the measures to take effect. Peter has proven in the past that he can handle such a difficult situation.
"Continuity in a football club is an important asset. Both myself and the Chairman had a long chat with Peter on Sunday, and he continues to be up for the fight and is desperate to succeed but understands results have to change, and quickly."