The Blues held a 4-1 lead after just 27 minutes before being pegged back by Slavia in the second half, but Maurizio Sarri's side survived the closing half-hour of the contest to prevail 5-3 on aggregate.
Chelsea will now take on Eintracht Frankfurt after the Bundesliga outfit recorded a 2-0 win over Benfica, a result which saw the Portuguese side bow out on away goals with the teams locked at 4-4 over the two legs.
The Premier League team went into the game knowing that avoiding defeat would secure their passage through to the next stage, but it quickly became apparent that Sarri had instructed his players to push for goals.
A second soon arrived in calamitous circumstances. Pedro could only strike the post from two yards out after meeting Eden Hazard's cross, but the ball ricocheted off the woodwork and was inadvertently diverted into the roof of the net by Slavia defender Simon Deli.
Slavia had been left stunned, and the Czech outfit went further behind after 17 minutes. Pedro was released clear on goal by a pass from N'Golo Kante, and the Spaniard unselfishly laid the ball into the path of Giroud to convert into an empty net from 15 yards out.
Chelsea had seemingly wrapped up the tie, but Slavia showed signs of life going forward with Deli seeing a header from a free kick superbly kept out by Kepa Arrizabalaga.
However, the goalkeeper was powerless to keep hold of his clean sheet as Tomas Soucek was left all on his own at the resulting corner to power a header into the net from 12 yards.
The visiting supporters finally had something to cheer in England's capital, but Chelsea soon reopened their three-goal advantage courtesy of Pedro scuffing a shot into the net after Kolar had denied Giroud.
The back-and-forth nature of the game subsided ahead of half time, but Slavia returned for the second half prepared to risk a heavier defeat by searching for the goals which they required to remain competitive.
Jindrich Trpisovsky's approach paid off within six minutes of the restart, with Petr Sevcik taking advantage of being provided with space on the edge of the area to lash a low effort into the bottom corner.
Chelsea had little reason for alarm with Slavia still requiring three goals, but the mood around Stamford Bridge changed for the worse as the visitors further reduced their deficit.
Sevcik was again provided with room to tee up a shot from 25 yards, and the winger proceeded to send a swerving effort into the far top corner of Arrizabalaga's net.
Chelsea had been left shellshocked with Sarri also becoming increasingly agitated on the bench, but that did not stop the Italian from making the decision to withdraw Hazard.
Slavia continued to plough forward with the aim of pulling off a miracle-like comeback, and there were signs that Chelsea were beginning to become wary of giving up such a commanding position.
However, the chances dried up for both teams during a relatively uneventful final quarter, much to the relief of Sarri who would have previously envisaged a routine victory after a hectic period for his squad.
Sarri's attention will turn to facing Burnley in the Premier League on Easter Monday, but the Italian will be aware that his side cannot afford more lapses in concentration if they are to finish in the top four in England's top flight and maintain their challenge in this competition.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Christensen, Luiz, Palmieri; Kante, Kovacic, Barkley (Jorginho 70'); Pedro (Hudson-Odoi 87'), Giroud, Hazard (Willian 65')
SLAVIA PRAGUE (4-2-3-1): Kolar; Kudela, Ngadeu-Ngadjui, Deli, Boril; Kral, Soucek; Sevcik (Stoch 78'), Traore, Zmrhal (Skoda 85'); Masopust (Olayinka 52')