Some started booing the moment the silence commenced before last night's match against Dagenham & Redbridge at the Racecourse Ground.
The booing, and some chanting during the 60-second period, led to other supporters shouting angrily at those causing the disturbance to quieten down.
On Wednesday morning Wrexham AFC said some were asked to leave the ground for their own safety.
In a statement the club said: "It was disappointing to hear the one-minute silence before the 4-1 victory against Dagenham & Redbridge disturbed be a small number of individuals among the 9,835 in attendance.
"To have the Club’s name in the media this morning for all the wrong reasons, is not a scenario that should have occurred and we wish to apologise to all who were offended by the actions of the minority, while thanking the vast majority who did observe the one-minutes silence.
"A number of the individuals who disturbed the one-minute silence were asked to leave the Racecourse Ground for their own safety."
Other supporters were reported to have stayed on the concourse until the one-minute silence was over before taking their seats.
One fan who was at the game said: "It sounded as though a relatively small number of fans decided to boo during the minute's silence. A couple of others responded to this by shouting that those responsible for the boos should show some respect.
"It was disappointing that a very small minority chose to behave in this way, but it's good to see the club has acted appropriately. It is not fair on the club to have its reputation tarnished by a small number of people who do not know how to behave."
One-minute silences were held at football grounds across England and Wales last night as EFL and non-league matches resumed, following the decision to postpone fixtures at the weekend.
Before the match Wrexham AFC asked fans to respect the silence.
"We have a simple request, regardless of your own personal opinions, which we recognise everyone is entitled to; on this occasion, you leave these at the turnstiles of the Racecourse Ground and do not use the one-minute silence to promote your own views over those of others," the club said on Tuesday before the match.
"The spotlight on the Club, shines brighter than ever and in an exceptionally positive way, so let’s not give anyone a reason to view the Club in any other way, by not observing the one-minute silence.
"The Club hopes it can count on your support in observing the one-minute silence."
Wrexham AFC's 4-1 win leaves them second in the National League, one point behind leaders Chesterfield after eight games.