Hakan Calhanoglu put the hosts in front, but Julian Brandt levelled somewhat fortunately for Leverkusen just before the hour mark.
With eight minutes remaining, Heiko Westermann volleyed in a winner to lift the northern Germany side above Stuttgart and Nurnberg ahead of their games this weekend, leaving Leverkusen at risk of losing fourth spot, and Hyypia at risk of losing his job.
Calhanoglu has already shown his ability to score spectacular goals with a direct free-kick from more than 45 yards against Borussia Dortmund earlier this year, and his opener in the fourth minute as another goal to savour.
Rafael van der Vaart touched the ball down for him to drill into bottom left-hand corner.
There were further chances at both ends in an open game which could have seen far more than just one goal come the interval.
Stefan Kiessling was denied by Rene Adler before the Germany striker headed another good opportunity wide.
Heung-Min Son also brought a good save out of Adler while Van der Vaart and Calhanoglu posed the greatest threats to Bernd Leno's goal.
Early in the second half, another spectacular Calhanoglu shot on the volley went dipping towards the top left-hand corner before it was pushed over the bar by Leno.
Against the run of play, Leverkusen levelled thanks to a favour from their former goalkeeper Adler in the form of an embarrassing blunder to fumble Brandt's shot into his own goal
It was the 18-year-old's first goal in the Bundesliga, but an unexpected one with Germany international Adler failing to get his full body behind what seemed like an easy save.
At this stage, Leverkusen appeared happier to settle for a point, but Hamburg had other ideas.
Eight minutes from time, Westermann had the HSV-Arena faithful on their feet celebrating what could prove to be a crucial goal.
The way he scored only his second goal of the season suggested the Germany centre-back should perhaps reconsider his true position on the field.
He met Dennis Diekmeier's cross from the right with a fine right-foot volley from near the penalty area which flashed past Bernd Leno before the Leverkusen goalkeeper could react.
It was a strike worthy of winning any game and provided three precious points for Hamburg, who survived a late scare when Emre Can was denied by Adler from close range.