With League leaders Yeovil in FA Cup action at Premiership Charlton, the Swans' win means they trail the Glovers by just one point.
Southend moved into contention for the automatic promotion places with a 2-0 win over Darlington, thanks to a 77th-minute goal from Mark Bentley and a stoppage time effort from Freddy Eastwood.
An 89th minute goal from Jonathan Parkin earned fifth-placed Macclesfield a 2-1 win over Bury.
Lee Unsworth had given the Shakers the lead after 14 minutes, with Parkin hitting the winner for the Silkmen after Michael Welch had levelled.
Cambridge's woes continued as they lost 3-1 at home to fellow strugglers Kidderminster despite playing for over an hour against 10 men.
Shane Tudor had given the Football League's worst side the lead in the 16th minute and, when Tom Bennett was sent off after 25 minutes, the chances of a comeback from the second bottom Harriers looked slim.
However, Chris Beardsley levelled for Kidderminster just five minutes later and second-half goals from Johnny Mullins and Gary Birch left Cambridge empty-handed.
Goals from Ciaran Toner and Simon Yeo ensured a 2-0 win for Lincoln over Mansfield, who had Luke Dimech sent off after 65 minutes.