Draper, who is bidding for a first title on the ATP Tour, enjoyed a confident start as he broke early to take a 2-0 lead.
However, Bublik broke back immediately and the pair were evenly matched with the world number 31 saving a set point at 6-5 before forcing a tie-break.
The British number four claimed a mini-break to go 4-2 up before closing out a gruelling hour-long first set in hot conditions.
Bublik entertained the crowd with dazzling drop shots as the players exchanged holds at the start of the second set, but he became frustrated when Draper broke and consolidated for a 4-2 lead.
Draper maintained his advantage and needed only the first of three match points to set up a final with world number 32 Lehecka, who thrashed American third seed Sebastian Korda 6-2 6-1 earlier on Friday.
“I’m very proud of myself. It was a tough match against a guy who is very unpredictable,” Draper said in his on-court interview.
“It’s really fun to play him and to watch him play as well. It’s not a very good mix when you’ve got a guy serving 136 miles per hour and you can’t see the ball because the sun is coming on to the court, but it was a really good match.”
Saturday’s final comes before Draper takes on American Marcos Giron in the first round of the Australian Open, which gets under way in Melbourne on Sunday.
Asked what it would mean to win his first ATP title, Draper said: “It would mean a lot to me. I came here, I made the semi-final – I have gone one further this year. I made my first final on the back end of last year and I’m in the final again so I am really hoping I can get it over the line.”
