Ramsdale, who was involved in an incident with Spurs striker Richarlison moments earlier, was collecting his goalkeeper gloves from behind the goal after the final whistle when a fan appeared to kick out at him.
A melee ensued with several players from both teams joining in, but Arsenal boss Mikel Arteta quickly ushered his players away, including Ramsdale, to celebrate in front of their own fans.
"The Spurs fans gave me some throughout the game," Ramsdale said postmatch. "I was giving some back. The people I did give it to greeted it in sportsmanlike terms, but one fan tried to give me a little punch on the back.
"It is a shame as it is just a game of football at the end of the day. Both sets of players tried to bring me away, but thankfully nothing too drastic happened. It is a sour taste, but I'm sure we will enjoy it when we go back into the dressing room."
Tottenham said they will work with local authorities and Arsenal to help identify the fan who kicked out at Ramsdale and are ready to take action once they have.
"The club has reviewed its CCTV footage to identify the supporter and will be working with the Met Police, Arsenal and Aaron Ramsdale to take the strongest possible action, including an immediate ban from Tottenham Hotspur Stadium," Spurs said in a statement after the match.
The Professional Footballers' Association released a statement post-game, saying: "Violence towards players is completely unacceptable. These types of incidents are happening far too often.
"Players have a right to be safe in their place of work. When a player is attacked, we expect the laws and regulations that are in place to protect players to be properly enforced.
"As the players' union, we treat this as a priority issue. We will continue working with the authorities to demand that players and staff are better protected in their place of work."
The FA echoed the sentiments of the PFA in their statement shortly after: "We strongly condemn the incident involving a spectator that followed today's Premier League fixture between Tottenham and Arsenal. This is wholly unacceptable behaviour and we will work together with the police, the relevant authorities and the clubs to ensure the appropriate action is taken."
Meanwhile, Spurs' boss Antonio Conte was not pleased with the action of the fan, calling on those in the stands to have respect for the players on the field.
"In every moment we have to show respect," Conte said. "If we want respect, we have to show respect. Honestly, I didn't see this. But if it happened, this type of situation for sure, I didn't like it."
The incident came after Premier League leaders Arsenal put on a masterclass to overwhelm Tottenham and extend the gap at the top to eight points. Ramsdale had been a key figure in the victory, denying Harry Kane on several occasions.
"The manager also mentioned that in the last few North London derbies we haven't kept a clean sheet, so that spurred me on a bit," Ramsdale added. "We deserved the two-goal lead and maybe deserved more."