Keith Curle's side had given themselves a mountain to climb after going in at the interval 2-0 down, but their second-half showing gave them a well-earned share of the spoils.
The visitors had started well and went close inside the third minute when Paul Mayo struck a left-footed effort narrowly wide after the Chester defence had only partially cleared a corner.
Chester responded well and on ten minutes Branch forced keeper Alan Marriott into a full-length save from his sweetly-hit strike on the edge of the box.
Referee Russell Booth did not help the flow of the game with his whistle-happy approach bringing several unnecessary halts to the play.
The Red Imps took the lead seven minutes before the interval when Gary Birch held off Phil Bolland and hit a superb pass to Francis Green. The midfielder made excellent progress down the left flank and crossed to Nat Brown who powered a header past Chris MacKenzie.
Four minutes later the advantage was doubled as a free-kick was missed by the entire Chester defence and the unmarked Birch superbly volleyed the ball home.
Within two minutes of the restart Chester almost pulled a goal back as Branch cannoned a shot off the crossbar.
On 50 minutes their early second-half pressure paid off when Ben Davies pulled a goal back. Ryan Lowe hung a cross in the Lincoln penalty area and Marriott should have claimed the bal but his handling error presented it to Davies who had the simple task of tapping home.
Carl Regan wasted a good chance to level matters, and after being set up by Branch he shot straight at the keeper.
The visitors appeared to have held out until Lowe was brought down in the box by Paul Morgan and Branch dispatched the kick, sending Marriott completely the wrong way.
To add insult to injury, the visiting captain was also sent from the field, seemingly after speaking out of turn to the official.