Yes - it can be done.

[Spiffy]

June 2016
S.A. v Ireland, Cape Town
Ireland's Stander red carded around 20 min, the team down to 14 men.
A total hammering by the gigantic Saffer monsters in sight. A belligerent Bok blitzkrieg beckoning.
Result : Ireland 26 - 20 SA. Orchestrated by P. Jackson. No Sexton.
A relentless and inspiring display of grit, determination, pride, passion, fire, bloody-mindedness, character, esprit de corps, might, power, physicality, muscularity, strength, athleticism, domination, total commitment, fitness, cohesion, skill, pace, coaching, planning, selection, tactics, whooping, hollering, chest thumping, head tapping, bum patting, eye-bugging, hugging, jig-dancing, buddy bonding etc.. and they actually won (and crossed the try line.)
A defeat of 15 by 14 at the international level - what a performance. And away from home too.
Who would have thought it possible?

[Sandydragon]

Being a player down can make some teams play better. Wales nearly beat France at the 2011 RWC and as you demonstrate, Ireland did beat the Bokke despite being down to 14 for much of the game.

Makes a mockery of the argument that an early red card ruins the game and should be avoided.