Draper’s run to the fourth round of the US Open earned him not just a second Great Britain call-up but a first appearance, with Smith picking him ahead of his top-ranked player Cameron Norrie and former world number one Andy Murray.
The 21-year-old fully justified his captain’s faith, thrilling a 9,000-strong crowd at the AO Arena by breaking Kokkinakis when he served for the match then coming from 4-2 down in the deciding tie-break to win 6-7 (6) 6-3 7-6 (4) after two hours and 52 minutes.
Evans then took to the court against world number 12 Alex De Minaur, the highest-ranked player in the four-team group.
De Minaur has had a brilliant summer but Evans is also in form having won the biggest title of his career in Washington and then reaching the third round of the US Open where he took a set off Carlos Alcaraz.
And he extended his tour-level winning record against De Minaur to 3-0, catching the Australian cold and then recovering from losing the second set to lead 4-1 in the decider.
There were some late nerves as De Minaur responded with two games in a row but Evans got his tactics spot on to complete a 6-1 2-6 6-4 victory.
That gave Britain an unassailable lead against last year’s runners-up, with a top-two spot in the group needed to secure progress to the quarter-finals and with matches against Switzerland and France still to come this week.
After drilling a final backhand winner down the line, Draper pretended to hit the bullseye, a reference to the competitive games of darts that have been keeping the team busy away from the court.
Speaking on court, he said: “There’s nothing better. It was a real battle, massive crowd in here. It’s amazing to play my first Davis Cup tie in the UK in this sort of arena. I’m just so happy to be here and grateful Leon trusted me and put me out here today.”
For most of his long tenure, Smith’s team, based around Murray, virtually picked itself. Greater options have left him with more difficult decisions and he was criticised for his selection at the same stage last year, when Britain made an early exit.
There will have been great satisfaction for the Scot, therefore, in the performance of Draper, who has been kept off court for much of the season by a string of frustrating injuries but carried the confidence of his run in New York into this clash against another 6ft 4in heavyweight in Kokkinakis.
There was very little to choose between them in the tightest of first sets, with Kokkinakis saving a set point at 4-5 and then clinching his second set point in the tie-break.
Draper came up with the perfect response, breaking Kokkinakis immediately to start the second set and looked to be in ascendancy at the start of the decider.
He could not get the break, though, and was in big trouble when he dropped serve at 4-4 before staging a rousing comeback.
Of his selection, Draper said: “Leon told me a couple of days ago. He said he wanted me to be out there and that he believed in me. I prepared well, trained well and only really started thinking about it an hour before I played, when the nerves started coming in.
“I knew I’d played some tough matches at the US Open and I felt really good about my tennis. That helped the nerves a lot. When I got out there with the home crowd and all those people supporting me, it felt amazing. I haven’t played too many great matches this season but I think that was one of them.”
Like Draper, 27-year-old Kokkinakis has been badly affected by injuries through his career, while he has had trouble closing out matches, including in a near six-hour epic against Murray at the Australian Open in January that finished after 4am.
He was left cursing a similar pattern here, saying: “I let my nerves get to me a little bit. Hats off to him, he played some good tennis when he needed to but it’s definitely a tough one, it stings for sure.”