Bjorn added to his title from 2011 after watching Lee's attempt from 15 feet slide left of the hole on the par-4 18th.
Earlier, both players missed 20-foot putts for the win in regulation on the foggy, rain-swept final green following a 30-minute suspension for fog to clear.
Bjorn, who waited out the delay standing on the 18th tee, missed left and completed his round of 6-under 65. The Dane then watched on TV in the scorer's hut as the 36-year-old Lee of Scotland sought to clinch his first career title.
Lee, who led by two shots after his 61 on Saturday, had his putt catch the right lip of the hole and carded a 67. Both finished with 20-under totals of 264.
Victor Dubuisson of France was third, one shot back after shooting 66.
Bjorn's 14th career European Tour title was his first since winning here in 2011, when he closed with 62 to also finish 20 under.
The 42-year-old Dane earned prize money of 366,660 euros ($483,000). Lee's career-high check for 244,440 euros ($322,000) looks certain to secure his European Tour status for next year.
Just like two years ago, Bjorn made a Sunday charge for the title in the thin mountain air of the Swiss Alps.
Bjorn fired five birdies in the first eight holes, and added another at the par-5 15th in his bogey-free round.
Indeed, his entire tournament was a model of consistency with just two dropped shots and no eagles in four rounds.
Bjorn birdied the first and fourth holes Sunday, and moved into a three-way share of the lead with another at the par-4 fifth. He joined Lee and Alejandro Canizares of Spain, who eagled the easy par-5 first and birdied the next two holes.
Reaching the turn in 31, Bjorn led by two but missed mid-range birdie chances at holes 11 and 12, as Lee made birdie at the par-4 12th.
Bjorn found trouble at the par-5 14th, hitting his tee shot onto the 13th tee box. He got a free drop and sank a 6-foot putt for par.
Lee drew level at 19 under with a 25-footer for birdie at the 14th, then both contenders got birdies at the 15th.
Leading scores after the fourth and final round from the EPGA European Masters at Crans-sur-Sierre on Sunday (GBR and IRL unless noted - Par 71):
264 - Thomas Bjorn (DEN) 66-66-67-65, 264 - Craig Lee 71-65-61-67 - Bjorn won at first play-off hole
265 - Victor Dubuisson (FRA) 68-65-66-66
266 - Alejandro Canizares (ESP) 69-65-65-67
269 - Ross Fisher 71-69-63-66, Miguel Angel Jimenez (ESP) 65-68-69-67
270 - Gregory Havret (FRA) 68-69-70-63, Brooks Koepka (USA) 68-66-69-67
272 - Thongchai Jaidee (THA) 68-69-70-65, Stephen Gallacher 67-65-72-68, Tommy Fleetwood 65-68-69-70
273 - Paul Lawrie 68-70-70-65
274 - Richard Sterne (RSA) - 69-66-71-68, Maximilian Kieffer (GER) 68-69-69-68
275 - Rikard Karlberg (SWE) 69-69-71-66, Tom Lewis 71-70-67-67, Andrew Dodt (AUS) 69-72-66-68, Brett Rumford (AUS) 68-68-70-69, Paul Casey 65-69-71-70
276 - Marcus Fraser (AUS) 69-71-68-68, Mark Tullo (CHI) 68-66-73-69
277 - Gareth Maybin 70-69-70-68, Angelo Que (PHI) 70-70-68-69, Simon Dyson 71-69-68-69, Jaco Van Zyl (RSA) 68-68-73-68