The Frenchman missed a chance to open the scoring early on, but did manage to find the back of the net after 10 minutes, heading in from a Joshua King cross.
Gestede claimed his second after 20 minutes with another header, nodding in from close range following a free kick from Craig Conway.
The visitors' high-tempo style continued to frustrate the Birmingham defence as Todd Kane found his way through just four minutes later to add a third Blackburn goal with a deflected finish.
Gestede completed his hat-trick deep in stoppage time in the first half with another header, resulting in the home supporters booing as their players left the pitch at the break.
Nikola Zigic picked out Federico Macheda just a couple of minutes after the restart as the hosts tried to get themselves back into the match, but Paul Robinson made a save with his legs to keep the ball out.
Zigic managed to find the target himself shortly after the hour mark as he headed in from close range from Callum Reilly's cross.
Conway came close to clinching a fifth for the visitors, but his 35-yard free kick just missed the left post.
The home side's hopes of an unlikely comeback were boosted inside the final 10 minutes when Demarai Gray rounded the defender on the edge of the box before firing into the bottom corner to score his first senior goal for the club.
Reilly tested Robinson during stoppage time as Birmingham tried to continue their comeback, but they were unable to reduce Blackburn's lead further.
The result leaves Birmingham just outside the relegation zone on goal difference, while Blackburn move up to ninth.