Croatian midfielder Luka Modrić scored a spectacular opener to hand Spurs a 37th-minute lead but Bolton equalised six minutes after half-time through Nigel Reo-Coker.
Rafael van der Vaart restored Spurs’ lead with a close-range finish and two quick-fire strikes from Emmanuel Adebayor sealed Tottenham’s first away win since beating Noriwch at Carrow Road in December.
Spurs’ victory moves them to within a point of third-placed Arsenal but Redknapp insisted his side would settle for fourth place.
“Anything can happen still, it is all to play for,” said Redknapp. “We have got to try to win the last two games and see where it takes us.
“We have still got to get fourth, never mind third.”
It was an emotional night for Bolton and Tottenham, with Fabrice Muamba returning to the Reebok for the first time since suffering a cardiac arrest during their FA Cup tie in March.
Redknapp praised his side for coping with an electric atmosphere and Bolton’s mini comeback after the interval following a disappointing first-half display by the Trotters.
“It was an important game for us and I thought we played ever so well in the first half – we dominated it from start to finish,” Redknapp added.
“We came in only 1-0 up, could have been two or three.
“But credit to Bolton, they came out second half and came at us a bit more, got the ball forward and they got the goal.
“At that stage the crowd was with them and they suddenly had a great little spell.
“It looked like they were the more likely team but we then showed fantastic ability on the break, pace, and really punished them.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news