The Swede won the playoff at the $2.56 million European Tour event when he birdied the second extra hole – the par 4 18th – with his putt from 18 feet.
Hanson needed a victory to have a chance of making Colin Montgomerie’s European team for the Ryder Cup.
“Today was really hard,” Hanson said. “I was lucky in the end, after I got off to a bad start. It feels fantastic.”
The overnight leader by four strokes, Hanson had a disastrous front nine that included three straight bogeys and he added a double bogey on the 12th.
But birdies on 5th, 10th and 16th brought him back into contention and Hanson closed with a fourth round 74 to finish with a 10-under 278 total at the Prosper Golf Resort.
“I put myself in a great position yesterday and I’m feeling absolutely great that I managed to do it,” Hanson said.
Irishman Lawrie, who started eight shots behind Hanson, produced an eagle on the 10th and closed with a 6-under 66, while English rookie Boyd ended with 68.
Englishman Simon Dyson – another Ryder Cup hopeful who needed a win to qualify – carded 70 and dropped to a share of fifth from second overnight.
Miguel Angel Jimenez of Spain, who co-designed the 7,155-yard course at the northeastern Czech town, shot a disappointing 73 to finish tied at seventh.
Jimenez was eighth in the standings before Celadna and close to automatically making the Ryder Cup.
Next week’s Johnnie Walker Championship at Gleneagles, Scotland, is the final event in the race for the nine automatic Europe Ryder Cup spots. Jimenez won’t be playing there.
“It means so much to get in the top nine,” Hanson said. “It turned out in the end that both me, Miguel and Simon Dyson, who had so much to play for, didn’t play our best in the end. But I managed to scrape it round the back nine and make that birdie on 16 to get into the playoff.”
The tour returned to the Czech Republic last year for the first time since 1997.
Sweden’s Oskar Henningsson won in Celadna in 2009.