By David Brenchley
The Toffees put in a strong performance and would have walked away with a point had it not been for the reflexes of Gunners keeper Emma Byrne, when she kept out Natasha Dowie's close range effort.
Taking the place of manager Mo Marley in doing post-match interviews, Spence said: "Overall, we were disappointed with [yesterday's] result but certainly not disappointed with the performance."
"I thought we dominated in the first half and we were actually a little bit disappointed to go in 0-0. We felt we should have been ahead.
"Second half we expected a response from Arsenal, they did that and probably the only time we let Kim get away she showed that quality from 20 yards and she puts the ball in the back of the net."
With just one league game remaining, the Toffees are now targeting securing their only piece of silverware in the Continental Cup, which starts on Sunday 4 September and could see the Merseysiders facing Chelsea two weeks in a row.
"First half of the season we had too many draws and that is what has ultimately cost us," Spence added. "The run we have gone on in the second half of the season has been fantastic.
"Some of the performances have been fantastic and to come to a place like Arsenal and dominate the game for long periods shows how far we have come. We just have to use that now to hopefully go into the Continental Cup and do well in that.
"We will see the Chelsea game as an opportunity for players to stake their claim for the cup competition coming up.
"We set ourselves the target of wanting to win silverware. The league is obviously gone so the next target is the Continental Cup."
photo Cheltenham jockey club
photo Cheltenham Racecourse
Slovakian alpine skier narrowly beats Millie Knight to capture gold photo Getty Images