Leverkusen conceded in the opening 15 minutes but fought back to win 2-1 at the BayArena, a result that saw them consolidate third position on the Bundesliga table.
Stuttgart took the lead on 12 minutes thanks to Vedad Ibisevic's penalty after Philipp Wollscheid fouled the Bosnian striker.
The home side responded in the second half, scoring twice in the last eight minutes through Stefan Kiessling and Lars Bender to claim all three points.
"Stuttgart has made it very difficult for us today," Hyypia said.
"Especially in the first half their defence was very good and we didn't play quickly enough and not precisely enough to have any good chances.
"Then we did a few changes at half-time and then in the second half we had more room to attack, but not so many positions to score. But then a football match lasts 90 minutes.
"And we also knew that Stuttgart has played a lot and has to play a lot after this match, and that we needed to keep the tempo high and try everything until the 90th minute to score.
"We did everything and of course we are happy, I am very happy for the team, that they showed good character today and in the end also won the game."
Leverkusen's victory moved them seven points clear of fourth-placed Eintracht Frankfurt, who suffered a 1-0 defeat at the hands of Borussia Moenchengladbach on Friday.