The Blues started with N'Golo Kante and Eden Hazard on the bench and were outplayed for long periods until Alonso headed in an unlikely winner four minutes from time.
Victory was scarcely deserved for Chelsea, with Kepa Arrizabalaga starring in the away goal and the notion of a kind draw for the visitors put to bed by an excellent Slavia display.
But Maurizio Sarri's decision to rest key men was ultimately justified as his side prepare to visit Liverpool on Sunday in complete control of this European tie ahead of next week's return leg.
An off-balance Simon Deli directed his second-minute header wide and Slavia continued to put Sarri's men under pressure.
Willian almost stole the opener against the run of play, crashing a superb 20-yard strike off the crossbar, but the hosts were well on top once more by the interval.
Hazard was called from the bench after almost an hour of Chelsea toil and they immediately clicked into gear, with Ondrej Kolar blocking well from Antonio Rudiger before Willian drilled just wide.
The Blues left spaces at the back to exploit, though, and only a smart Kepa parry denied Ibrahim Traore a breakaway winner, while the goalkeeper later kept out an acrobatic Jan Boril effort.
And those saves gave Chelsea the foundation to chase a vital goal that arrived as Willian's deep cross picked out Alonso for a powerful downward header.
What does it mean? Chelsea escape
Chelsea might have taken a draw before Thursday's match - they surely would have done after 85 gruelling minutes. But Alonso's goal changes everything. Each of the Blues' attacks were against the run of play, but they had more cutting edge than Slavia and look set to progress for that reason.
Alonso back at it
Alonso regularly scored vital goals for Antonio Conte's Chelsea two seasons ago, but he has not always been a guaranteed starter for Sarri in 2019. Afforded an opportunity in Prague, he came up with the goods. Few full-backs manage to take up and deliver from such threatening attacking positions.
Slavia's Stoch sub
As former Chelsea man Miroslav Stoch ran the show in the first half, it appeared as though Sarri's selection calls would be those questioned. But Jindrich Trpisovsky's decision to hook Stoch 25 minutes from time turned the tie. Slavia no longer had control, even if they kept posing questions, and Chelsea got the winner.
Chelsea return to Premier League action against title-chasing Liverpool on Sunday, while Slavia face rivals Sparta Prague in a derby clash. The sides will then head to Stamford Bridge for next week's second leg.