Harry Kane's second-half header secured the 1-0 win for Spurs at Wembley, which leaves the Gunners six points adrift of fourth-placed Liverpool, having played a game more.
"It was a game we couldn't afford to lose," he told a news conference. "It makes it more difficult now [to finish in the top four] but we have to fight.
"[Our away record] is very poor. I believe as well, once they scored, that played a part."
Wenger appeared frustrated that Kane escaped punishment for a push on Laurent Koscielny when he headed in Ben Davies' cross, but he later conceded the goal was fair.
"I have no complaint," he said. "I watched it again, it’s a regular goal. We can only look at ourselves.
"He is a super striker, one of the best in the world who scores against everyone."
Arsenal offered very little in attack for much of the match, with Pierre-Emerick Aubameyang, Henrikh Mkhitaryan and Mesut Ozil all disappointing, before substitute Alexandre Lacazette missed two clear chances in the closing minutes.
The Gunners have been criticised for their defensive displays at times this season but Wenger believes it is a failure to take their chances that has cost them most dearly.
"The game should have been finished at half-time for me. We missed opportunities on the counter-attack. That is not acceptable at this level," he fumed.
"The second half, we should have lost the game in the first few minutes, they had lots of chances. At the end, we should have come back to 1-1.
"Football is about scoring goals. You can say we are not strong enough, but you look at the numbers and we don't score enough goals.
"I believe that we had opportunities, dangerous situations that we didn't transform into opportunities. That's where my regret was. In the final ball, we missed something.
"Overall, when you have two or three opportunities like we had in the final part of the game, you have to take your chances. You will not have 10 chances; if you don't take them, you will suffer at the end of the day."