City started brightly and Scott Shearer was forced to save from Derek Asamoah in the second minute, but Rushden retaliated five minutes later when Alan Marriott was forced to make a fine save from Neil McCafferty's free-kick.
A minute later Billy Sharp should have done better when, unmarked, he headed straight at Marriott.
After 14 minutes Rushden took the lead when Broughton found Sharp, who cleverly beat two men, before shooting past Marriott from 15 yards out.
Six minutes later Lincoln equalised when McCafferty missed his tackle in midfield and Peter Gain found Simon Yeo, who hit an unstoppable shot from the edge of the box that left Shearer helpless.
Broughton should have scored for the home side after 25 minutes but headed wide following a fine cross from Sean Connelly.
In the second half both sides found it hard to create clear-cut chances, although Sharp went close in the 61st minute when he turned superbly and volleyed inches wide.
Broughton was dismissed in the 73rd minute when he was adjudged to have stamped on a Lincoln defender and Rushden paid dearly when Ben Futcher scored from close range after the home defence failed to clear a free-kick.
Lincoln were in control and after 79 minutes substitute Gary Taylor-Fletcher found Jamie McCombe who finished brilliantly.
City put the game beyond doubt when Taylor-Fletcher finished off a fine move seven minutes from time.
Rushden should then have pulled a goal back in the 90th minute but Stuart Gray shot straight at Marriott.