Nielsen made the vital breakthrough just seconds before half-time, sweeping home Clive Wilson's cross after a surging run by David Ginola and then spent a spell in the second half searching for a missing contact lens which he eventually found.
But Bolton, now 12 games without a win, found themselves still deep in trouble near the foot of the table after a frenetic struggle at White Hart Lane in which they called into question the eyesight of the referee, Peter Jones, after only 90 seconds.
Spurs keeper Espen Bardsen survived a furious Bolton penalty appeal. Dean Holdsworth went tumbling in the penalty area after Bardsen rushed from his line trying to repair a sloppy piece of defending by Wilson, but Mr Jones ruled there had been no contact and brushed off the protests of Bolton players. Bardsen then produced a wonderful second-half save to keep out a Jimmy Phillips shot
Bolton tempers overflowed in a spate of bookings for Nathan Blake, Holdsworth and the former Spurs player Chris Fairclough.
But what really finished them off was that sensational save by Bardsen, 15 minutes from time, when the Spurs reserve keeper stretched every sinew of his 6ft 5in frame to turn away a curling effort by defender Phillips.
Jurgen Klinsmann's return to the Spurs attack, just four weeks after breaking his jaw at Barnsley in the FA Cup, provided a tonic for the London side and he looked sharper than in any of the seven previous games before the injury, in which he had scored only one goal.
But Klinsmann should have doubled his tally instead of hooking high over the bar from close range in the 34th minute after a spectacular miskick by Pollock had sent the ball spiralling back into the danger area.
But bang on half-time the busy Nielsen, back in the team and establishing himself again after breaking a bone in a foot earlier this season, made no mistake when sweeping Wilson's low cross into the far corner after a typical burst of genius by Ginola had paved the way.
The Frenchman continued to inspire Spurs with his individual skill, but was too often guilty of overelaborating and there was a blow for Christian Gross' team when Chris Armstrong, trying to bundle home Sol Campbell's powerful header, collided with Keith Branagan and had to be carried off to rejoin the lengthy Spurs injury list.
However, a brilliant double-handed stop by the Irish keeper prevented Ruel Fox closing up the game for Spurs and Ginola might have twice added to the tally before being replaced by pounds 2.3m new signing Moussa Saib in the closing minutes.
Goal: Nielsen (45) 1-0.
Tottenham Hotspur: Bardsen; Calderwood, Fox (Brady, 87), Nielsen, Armstrong (Howells, 54), Carr, Ginola (Saib, 85), Wilson, Campbell, Klinsmann, Berti. Substitutes not used: Mabbutt, Grodas (gk).
Bolton Wanderers: Branagan; Cox, Frandsen, Fairclough (Johansen, 80), Pollock (Gunnlaugsson, 69), Blake, Thompson, Bergsson, Todd, Phillips, Holdsworth. Substitutes not used: Sheridan, Strong, Ward (gk).
Referee: P Jones (Loughborough).
Bookings: Tottenham: Nielsen. Barnsley: Fairclough, Holdsworth, Blake.
Man of the match: Bardsen.
Attendance: 29,032.Reuse content