The Blues drew 2-2 at the Emirates but Morata fluffed his lines in front of goal, twice firing off target when clean through and then at Petr Cech in stoppage time.
Morata has 10 Premier League goals in 19 appearances since joining from Real Madrid last year, but after failing to find the net with simple chances against Stoke City in Chelsea's 5-0 win on Saturday he has not enjoyed the festive season.
"It is not a lucky period for him," Conte told a news conference. "In the last game against Stoke I said it was important for the striker to have chances to score.
"Today he had chances but this is football. I'm happy with his commitment, he worked hard for the team and they are the most important things for me.
"For the striker it's very important to score and in this case your confidence can go down. He's very young, he has to continue to work, continue to improve. The goal is coming."
Morata, 25, fired well wide when sent through on goal early in the match.
Arsenal took the lead through Jack Wilshere but Eden Hazard's penalty and Marcos Alonso put the Blues in front before Hector Bellerin equalised in stoppage time.
Morata should have given Chelsea a last-gasp winner but Cech saved his effort when he only had the goalkeeper to beat before Davide Zappacosta crashed a shot against the crossbar.
A disbelieving Conte beat the turf in frustration, admitting the wasted chances cost third-placed Chelsea as they try to catch runaway leaders Manchester City.
"To draw at the Emirates is a good result," added the Italian. "(But) at the end of the game if we see the chances we created to score, for sure I think we dropped two points."
(Reporting by Rik Sharma, editing by Ed Osmond)
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