Only two minutes had gone when Thomas Orr providing the pass for McKenna to slot home his ninth goal of the season.
Shire's Richard Barnard saved from Scott McLaughlin, but Clyde's Marc Walters also had good saves.
Both teams had chances in the second period, but Shire'w win narrowed the gap with Berwick Rangers to two points.
Meanwhile, Clyde missed an opportunity to draw level with third-top Queen's Park with a game in hand.
David McKenna's early goal gave bottom side East Stirlingshire a narrow win that denied promotion hopefuls Clyde a fourth straight League Two victory.