The visitors had led remarkably led for 96 minutes of the contest at the Toughsheet Community Stadium but ultimately had to settle for a point.
Draper got the opener with the first attack of the game and Bolton found an equaliser with virtually the last after Tom Hamer’s own goal.
Steven Schumacher made two changes from the draw against Reading in midweek, with skipper Eoin Toal and Josh Dacres-Cogley returning to the backline.
Ethan Erhahon got the nod against his former club, partnering Xavier Simons in the midfield department again.
Bolton had a nightmare start as Reeco Hackett got down the right and picked out Draper, who rose highest to nod home after only 20 seconds. Teddy Sharman-Lowe got a hand to it but couldn’t keep the ball out of his net.
It was the last thing Schumacher’s side needed after their late frustration against Reading earlier in the week, and there was perhaps an element of fatigue in the slow start due to the quick turnaround.
George Johnston then made an important clearance from Hamer’s header across goal as Lincoln put the Whites under early pressure.
At the other end, Amario Cozier-Duberry’s over-hit cross forced George Wickens into an awkward save to his right. Josh Dacres-Cogley then had a shot blocked and the rebound fell to Joel Randell, who was denied by a smart stop from close range.
The visitors continued to pose a threat from crosses and Draper had another sight at goal, but this time his scuffed effort was held by Sharman-Lowe.
The Chelsea loanee was called into action again midway through the first half to keep out Hamer’s strike after he found a pocket of space at the far post.
Ibrahim Cissoko looked lively as the half progressed and came close to finding a team-mate in the middle with a couple of dangerous crosses.
Just as it felt like Bolton were starting to get on top, they were almost hit with a sucker punch as Hackett whistled a strike just wide of the near post.
Schumacher opted a couple of changes at the interval, sending on Aaron Morley and Josh Sheehan for Erhahon and Simons, as Wanderers searched for a response in the second half.
Morley was inches away from finding an equaliser with a well-struck volley after good work by Max Conway, but the crossbar came to Lincoln’s rescue.
Thierry Gale also joined the action, in place of Cissoko, as the Whites aimed to turn up the pressure.
Cozier-Duberry thought he had found an equaliser just after the hour mark, turning home from close range, but Wanderers’ celebrations were dashed when the Brighton loanee was deemed to have handled.
Wickens then made another big save to deny Randall with his outstretched foot after a slick move involving Gale and Conway down the left.
Sam Dalby was introduced off the bench with just over 16 minutes of normal time remaining, partnering Burstow in attack.
Wanderers continued to knock on the door as Cozier-Duberry played in Burstow with a clever pass. His powerful strike was pushed into the air by Wickens and nearly crept in, but a defender managed to get back and clear off the line.
The Whites were then awarded a free kick on the edge of the box and Sheehan went for goal, but Wickens was once again equal to the danger.
Richard Taylor replaced Conway in the final minutes, making his debut after recovering from a groin injury picked up in pre-season.
It looked like Bolton’s dominance in the second half would go unrewarded until the final moments of seven added minutes when Cozier-Duberry got down the right and played a dangerous ball across goal. Hamer tried to clear under pressure from Dalby but could only force the ball over the line, sending the Toughsheet into jubilation.