Renee Slegers said Arsenal had only achieved "the bare minimum" after Saturday's 3-0 win over Aston Villa confirmed Champions League qualification for last season.
Arsenal's Women's Super League title hopes were ended on Wednesday following a 1-1 draw with Brighton, as Manchester City ended their 10-year wait to be champions.
But the Gunners regrouped for a comfortable win at Villa Park, which ensured they are guaranteed to finish in the top three.
Alessia Russo scored in either half and also turned provider for Frida Maanum's goal in between those strikes, as Arsenal moved five points clear of fourth-placed Manchester United – who only have one game remaining.
Slegers' side are one point behind second-placed Chelsea with a game in hand, and the Gunners boss is now targeting a runner-up finish.
"That's the bare minimum to achieve at Arsenal," Slegers told Sky Sports when asked about the importance of Champions League football.
"I am so happy we're there now, but we want more. We want to go for second.
"This was an important win today. It was our shortest turnaround of season as we played Wednesday night and then on Saturday morning again.
"It was really good of the squad to be this ready, to put on this professional performance, nullifying Villa's threats and creating a lot going forward, so I'm happy with that.
"We bounced back. In adversity and when things are challenging and hard, we step up. I see how our individuals did that today and I am so pleased."
After overcoming Villa's Natalia Arroyo at the third attempt (previously D1 L1), Slegers has defeated all 20 managers she has come up against in the Premier League.
The Gunners boss reserved particular praise for Russo, whose 17 goal involvements in the WSL this season (11 goals, six assists) are second only to Khadija Shaw (23).
"She is one of the players that will always step up, that wants more and to be better. She bounces back after challenges," Slegers said of Russo.
"She's been leading the squad, she has mastered the number nine and 10 roles and she brings goals, assists and so much more."
Arsenal's final two WSL games are at home to Everton on Wednesday, then away to Liverpool on May 16.