Leading 1-0 from last week's first leg in the Czech capital, Chelsea appeared to be cruising towards the last four after an early goal blitz at Stamford Bridge.
Pedro struck the opener on five minutes before an unfortunate Simon Deli turned into his own net and Olivier Giroud added a third with barely quarter of an hour gone in London.
Tomas Soucek pulled one back for Slavia on 26 minutes, but Spain forward Pedro replied immediately to restore the Blues' three-goal lead.
Maurizio Sarri's side were made to sweat though as two sublime, quick-fire goals by Petr Sevcik just after half-time gave Slavia hope of an incredible comeback with 35 minutes still to play.
Chelsea eventually saw off the threat to progress 5-3 on aggregate despite an unconvincing second-half display.
"We have been efficient up front and solid at the back but we dropped our intensity after the break," Giroud, who scored his competition-leading 10th goal, told BT Sport.
"We started very badly in the second half but I'm really, really happy because we played well in the first 45 minutes and we are in a European semi-final which is very important to us," added Chelsea boss Sarri.
The 2013 Europa League champions will meet Eintracht Frankfurt in the last four after the Germans eliminated Benfica on away goals.
- Lacazette on target -
Eintracht lost 4-2 in Portugal a week ago, but goals by Filip Kostic and former Bayern Munich midfielder Sebastian Rode sent the German Cup holders through at the Commerzbank Arena.
Kostic turned home a rebound on 37 minutes after Mijat Gacinovic's curling attempt cannoned back off the upright.
Rode, on loan from Borussia Dortmund, hit the decisive goal on 67 minutes as he picked out the bottom corner from the edge of the area.
In Italy, Alexandre Lacazette drilled in a terrific free-kick on 36 minutes at the Stadio San Paolo as the Gunners completed a 3-0 aggregate victory over Carlo Ancelotti's Napoli.
Petr Cech denied Jose Callejon with his legs following a rapid Napoli break on the quarter-hour as the Italians sought to overturn a two-goal deficit from the first leg.
But Lacazette's 30-yard drive all but put the tie beyond Napoli, shortly after Aaron Ramsey hobbled off with a hamstring injury -- potentially his final appearance for Arsenal ahead of a move to Juventus in July.
"We came here with ambition and wanted to win and score some goals. We did it with Laca," Arsenal captain Laurent Koscielny told BT Sport.
"We kept a clean sheet, which is important for the confidence. Each game we have in front of us is a final."
Arsenal boss Unai Emery will come up against one of his former clubs Valencia, the 2004 UEFA Cup winners, for a spot in the final.
Valencia eased through with a 5-1 aggregate win over Villarreal in an all-Spanish affair.
Toni Lato and Dani Parejo grabbed the goals in a 2-0 second-leg victory for Valencia coach Marcelino against his former club.
The semi-finals will be played over May 2 and 9, with this year's final in Baku on May 29.