Costa will likely depart Chelsea for good in the next week as transfer deadline looms after a summer of controversy which sees him in a self-imposed exile from the London club.
And, while Morata's display wasn't perfection, it forms part of a promising start to life in English football for the 24-year-old.
The Blues dominated throughout the match but were slow on the break, while Morata lost a few physical battles with Everton's bruising three-man defence.
However, the Spain international showed his impressive aerial prowess as he headed a deflection into the path of Cesc Fabregas who produced a stunning outside-of-the-boot finish to open the scoring.
The former Real Madrid striker then bagged a goal of his own when he got on the end of Cesar Azpilicueta's cross as Chelsea's Spanish contingent continue to conjure the goals for the Blues.
Antonio Conte's 3-4-3 formation ensures that Chelsea play with great width, but Fabregas and Azpilicueta are also a constant threat with their deep crosses, which Morata is great at feeding off.
The 24-year-old is dominant in the air and he will only improve as the season goes on, getting used to Conte's tactics as well as the physical nature of the league. He seemed to grow after his goal, but some of his link up play was still below what Costa was offering last season.
2 - Alvaro Morata is the first player in Premier League history to both score & assist a goal in each of his first two home games. Handy. pic.twitter.com/tbIJ8tzA5P— OptaJoe (@OptaJoe) August 27, 2017
Strikers are judged on goals and Morata has already got two goals and two assists at Stamford Bridge in his 109 minutes of Premier League football.
The imminent return of Eden Hazard should help him add to that tally as defences worry about the pace of the Belgian running in behind them. With Hazard and Morata, Chelsea will have the option to go direct with size and speed up front.
Costa won more points in the Premier League than any other player last season, but on Sunday Morata was involved in earning his side points for the first time. He will need to show consistency, but Chelsea looked like the champions of last season even without their absent striker.
Didier Drogba met Morata this week at Cobham Training Ground and he must feel that his club is in safe hands with their new number 9.
With six points secured from their first three games and, with further forays into the transfer market expected in the next couple of days, Chelsea look to have successfully navigated clear of crisis.