Bosch kept his composure to slot over a 43-metre kick after the clock had ticked past 80 minutes. Racing were trying to see out time while in possession but Fabrice Metz was penalised for going off his feet in a ruck.
Sarries will travel to France again for a last four meeting with Clermont Auvergne after the Top 14 side dismantled Northampton 37-5 on Saturday.
“I told him (Metz) many times to keep his feet on the ground,” Racing scrumhalf Maxime Machenaud who scored all his team's points with a try and two penalties, told BeIN Sport.
The other quarter-final of the day features holders Toulon against Wasps with the winner hosting Leinster in the last four.
In Colombes, Charlie Hodgson kicked an early penalty as Racing were sanctioned in the scrum but the hosts were the most aggressive side.
Saracens were reduced to 14 men when Jim Hamilton received a yellow card for collapsing a maul near his try-line and Racing quickly benefited with Machenaud diving over after the pack had come close following a lineout.
Hodgson then missed two penalties in quick succession before finding his range again early in the second half to put his side ahead.
Racing were then penalised for going off their feet at the ruck and fullback Alex Goode stretched the visitors' lead to four points.
Machenaud put his side ahead with a couple of penalties but another discipline error cost them dearly. – Reuters
Original source: Bosch sends Saracens into semis