Huddlestone netted the only goal of the game with a stunning 25-yard strike which helped Spurs stay in fourth place with just two games left to play.
Tottenham are just a point above Manchester City, but can guarantee a Champions League place for the first time in the club's history with victory at Eastlands on May 5.
Tottenham could have opened the scoring on 17 minutes when the ever impressive Gareth Bale fired a low shot from the edge of the box which produced a fine save from Jussi Jaaskelainen diving low to his right.
You could sense the nerves among the home supporters with so much at stake and Bolton almost capitalised on 26 minutes when Gretar Steinsson saw his shot take a wicked deflection off Michael Dawson as the ball flew inches past the post.
But Huddlestone gave the home fans something to shout about when he broke the deadlock in superb fashion on 38 minutes.
Tottenham fans have been spoilt for choice when it comes to stunning goals at the Lane this season - none more so than Danny Rose's spectacular volley against Arsenal in last month's north London derby.
But Huddlestone came close to equalling that when he latched onto a ball from Benoit Assou-Ekotto and fired an unstoppable 25-yard right-foot shot which sailed past Jaaskelainen into the roof of the net.
Tottenham started the second half in anxious mood and Bolton were certainly making the home crowd sweat as they forced three corners in succession and were determined to take something from the game and not just make up the numbers.
Huddlestone almost eased the tension with another spectacular strike on the hour mark, but this time his 25-yard fizzing shot went just wide of the post.
Heurelho Gomes came to Tottenham's rescue on 62 minutes when he made a fine one-handed save diving low to his left to palm away a shot from Matt Taylor.
Tottenham should have put the game beyond Bolton on 66 minutes when Bale broke free inside the box and saw his shot superbly saved by Jaaskelainen. The Finnish keeper just about did enough to push the ball past the post.
Bolton refused to lie down and it could have been a different story for the north London club had Taylor not seen his shot cleared off the line by Younes Kaboul on 67 minutes.
Redknapp knew he needed to change things around and brought on Peter Crouch and Aaron Lennon, who was making his first appearance at White Hart Lane since December after being forced to miss the last five months with a groin injury.
But it was Gomes who came to Tottenham's rescue once again on 87 minutes when he made a stunning save to deny Taylor's dipping volley from the edge of the box.
Tottenham held on and are now bracing themselves for one of the games of the season in Manchester on Wednesday night.