Martial passed a fitness test on Saturday morning but was replaced at half-time during United’s 2-1 win at Old Trafford.
The French striker has been plagued by injuries since the start of the season, and Ten Hag, who is already short of options up front, said he did not want to risk another spell on the sidelines.
“I hoped to avoid him getting injured but he was complaining,” Ten Hag said.
“That’s why he also didn’t train in the week and was a question mark [for the derby]. We decided, and he decided as well, to start. He begged to start.
“I knew the start would be so important in this game and he is really good at pressing and he is a really big element in our pressing. But I needed to re-evaluate at half-time as you saw he was not capable of going 100% and this is what you need.
“Also to avoid him getting injured so we took him off but it also had a negative impact on our game after half-time.”
New loan signing Wout Weghorst, who watched the derby from the stands, is available after training at Carrington on Monday and could start at Selhurst Park if Martial or Rashford is ruled out.
“I have to think about that,” Ten Hag said. “First settle down after this game, make the right conclusions and then we will make a clear plan for Palace.
“Of course I know a lot about Palace, we faced them also in preseason and know how their team is, but the total plan, I don’t have now. We can construct the plan and prepare the team in the right way.”
United travel to Palace on a high after their derby victory moved them to within one point of Pep Guardiola’s side.
Ten Hag encouraged his players to celebrate their win over City but urged the squad to re-focus quickly as they look to extend their winning run to 10 games. The last time they achieved a similar run was under Sir Alex Ferguson in 2009.
“I think every win you have to celebrate,” Ten Hag said. “You work so hard for such momentum and every win is so great in top football and if you can’t celebrate you kill the energy as well.
“So we have to celebrate but after we have to settle down after 24 hours and move on to the next game. You have to get in the right rhythm every third day and we have to deliver.
“The players have to get in the right rhythm from a physical point of view but also mentally — be ready for the next game.”