The Blues came into the game with a 1-0 lead and had won 11 out of 11 ties in Europe when they held a first leg advantage so the omens were very much on their side.
And so it proved when they raced into a three-goal lead in side 20 minutes thanks to goals from Pedro, a Simon Deli own goal and Olivier Giroud.
Tomas Soucek gave Slavia Prague a glimmer of hope when he pulled one back on 25 minutes, but Pedro grabbed his second of the game moments later to restore the three-goal cushion on the night.
However, two quick second half goals from Petr Sevcik put the tie back into the melting pot.
And, despite a barrage of pressure from the Czech outfit for the remainder of the game, Chelsea held on to book a two-legged last-four showdown against Eintracht Frankfurt who dumped out Benfica.
Chelsea boasted five changes to the starting XI that lost 2-0 at Liverpool on Sunday as Andreas Christensen, Ross Barkley, Mateo Kovacic, Pedro and Giroud replaced Antonio Rudiger, Jorginho, Ruben Loftus-Cheek, Callum Hudson-Odoi and Willian.
Chelsea started well but were given a moment of concern in the second minute when Eden Hazard went to ground having been caught on his Achilles.
The Belgium international was given some treatment before returning to the action much to the relief of Maurizio Sarri and his coaching staff.
The hosts responded in the ideal manner when they took the lead after five minutes when Giroud slipped the ball through to Pedro towards the near post and, while goalkeeper Ondrej Kolar got a hand to the Spaniard’s effort, he couldn’t keep it out of his net.
Pedro should have claimed his second on nine minutes when he was teed up by Hazard in front of goal but somehow managed to fire his shot against the post. But his blushes were spared as the ball rebounded back into the net of centre-back Deli’s head.
Slavia Prague tried to rally, but found themselves three-down on 17 minutes when N’Golo Kante’s defence-splitting pass played in Pedro who then teed up Giroud for a simple tap-in.
Kepa Arrizabalaga was called into action for the first time on 25 minutes when made a smart save with his legs to prevent Deli from scoring at the right end with a header from Jaromir Zmrhal’s free-kick.
However, the visitors did pull one back from the following corner when captain Soucek was left unmarked in the area and powered his header past Kepa.
Chelsea issued an almost immediate response on 27 minutes when Giroud saw his effort saved by Kolar, only for Pedro to scuff in his second of the game from close range.
With a 5-1 aggregate lead, Chelsea were guilty of taking their foot off the pedal and they were made to pay when Slavia Prague scored twice in the space of two minutes early in the second half.
On 52 minutes, Sevcik picked up possession on the right and cut inside before unleashing a fierce left-foot drive into Kepa’s bottom corner of the net.
And they were firmly back in the tie moments later when Sevcik produced another stunning strike as he picked up the ball just outside the box and arrowed his shot into the top corner.
Sevcik should have grabbed his hat-trick and equalised on the night on 62 minutes but slipped at the vital moment when he looked poised to slot the ball into the net at the far post.
Slavia Prague continued to look the more likely to score again, but Chelsea weathered the storm and booked their place in the semi-finals.