The game was heading for a draw when Jermain Defoe broke free in the box with just one minute left on the clock and was brought down by West Brom keeper Tomasz Kuszczak.
Keane kept his nerve to slot the spot-kick into the roof of the net for his 100th career league goal and ensured Spurs moved five points clear of arch rivals Arsenal in the table.
It was harsh justice on a West Brom side who are fighting to secure their Premiership status and who had taken a first-half lead through skipper Curtis Davies.
But Keane's double strike in the second half gave Martin Jol's side a genuine chance of cementing a top four finish come the end of the season, while West Brom will be looking anxiously over their shoulders in the final weeks of the campaign.
Nathan Ellington could have opened the scoring for the away side on 13 minutes when he found himself unmarked on the edge of the box.
But with too much time on his side, the former Wigan striker fired a 25-yard shot just past the post.
West Brom continued to dominate the opening exchanges and were given their just rewards on 20 minutes when skipper Davies broke the deadlock.
Jonathan Greening's curling free-kick into the box picked out Davies who rose above Michael Dawson to flash a close-range header into the corner of the net.
Tottenham had to wait until the 40 minute barrier to force their first attempt on goal.
Defoe held off the challenge of two West Brom defenders before striking a fierce drive from the edge of the box which produced a fine save from Kuszczak diving away to his right.
The home side finally started to realise they had a game on their hands and came close again just 60 seconds later when Michael Carrick's corner picked out Ledley King.
King's shot looked destined for the back of the net but Ronnie Wallwork made a fantastic block to help deflect the shot over the crossbar.
And teenage winger Aaron Lennon saw his powerful drive just pushed round the post by Kuszczak on the stroke of half-time.
Tottenham came out in the second half determined to get back on level terms and came close again on 53 minutes.
Midfielder Jermaine Jenas picked the ball up on the edge of the box before striking a powerful 25-yard shot which sailed just over the bar.
Kanu almost made a spectacular return to north London on 57 minutes when his 30-yard dipping drive beat Paul Robinson in the Spurs goal but the ball flew inches over.
Jol decided a change was needed for Tottenham on the hour mark and substitute Mido almost made an immediate impact with his first touch of the ball.
Defoe released the ball into the path of the Egyptian striker whose first time shot inside the box went inches past the post.
But the home side finally managed to find a way past Kuszczak on 68 minutes when Keane came to Tottenham's rescue once again.
Carrick showed once again why he is pushing hard for a place in the England side with a fantastic ball to set up Keane inside the box.
The Republic of Ireland international made no mistake lifting the ball over Kuszczak into the back of the net for his 12th goal of the season.
And there was late drama at White Hart Lane as Keane grabbed his second of the game on 89 minutes to secure a valuable three points for the home side.
Defoe was brought down inside the box by Kuszczak and, although Mido had slotted the ball into an empty net, referee Chris Foy had already pointed to the penalty spot.
But Keane fired the ball into the top left-hand corner to keep alive their dream of playing in the Champions League next season.