Keane pounced on 15 minutes to side-foot home and send Spurs through as two sides who had shared 14 goals in their previous two meetings could manage only a single strike between them this time.
Spurs bossed the first half as Reading, with a string of fringe players operating in a complicated formation, struggled to find their feet in the slippery conditions.
The home side's attack was spearheaded by Leroy Lita, who was given the chance to prove his worth having started only four Premier League games this season.
But his evening got off to a miserable start when he saw an early poked effort saved by Radek Cerny as Lita tumbled before being booked for diving.
Instead, it was Keane who broke the deadlock, following up after Younes Kaboul's effort had been brilliantly saved.
The goal was tough luck on Reading keeper Adam Federici, who had reacted superbly to push Kaboul's header onto the crossbar only to see the rebound fall kindly for Keane.
Federici produced more fantastic reflexes five minutes before half-time when he reacted superbly to beat away Jermain Defoe's thumping volley from a Kevin-Prince Boateng cross.
The save seemed to sting Reading into life and they produced their best spell of the half in the minutes before the interval, before re-emerging with a more promising 4-4-2 formation for the second half.
Twenty-year-old striker Simon Cox was introduced on the right-hand side of midfield at the expense of teenage defender Alex Pearce and the home side suddenly carried a threat.
Nicky Shorey bent a shot wide after a neat Reading build-up, though the left-back otherwise looked lost after being pushed into a central midfield role.
But when he was replaced by Stephen Hunt, who had scored twice in the first game between these two sides, Reading looked increasingly likely to force a way back.
Liam Rosenior blazed over after Hunt's cross had been blocked following a quick James Harper free-kick and Hunt himself was then desperately unlucky to see a curling effort smack against the crossbar after some flowing Reading football.
That, however, was as close as the home side came and Keane's early effort proved enough to secure Spurs a trip to Old Trafford in ten days time.