Sidebottom finished the day with figures of 5 for 37 off 14 overs, while Monty Panesar took 3 for 33.
The not out batsmen for the hosts are captain Daniel Vettori on 13 with Jeetan Patel on six.
"Pity we didn't get the last two wickets, it would have been nice to get another one tonight," Sidebottom told reporters.
"The first half-hour [on Sunday], we have to be right on the money.
|"Pity we didn't get the last two wickets, it would have been nice to get another one tonight."|
"Daniel Vettori will be a key wicket, he's in good nick, but if we can open up an end and get Chris Martin in, the onus is on us to play well."
Prior to Sidebottom's hat-trick, the Black Caps were batting well and were looking to set a target in excess of 300 for England to chase on the final day on a pitch that is producing noticeable turn.
After Sidebottom had opener Matthew Bell out caught behind for a duck on the third ball of the innings, Fleming and Jamie How put on a 98-run second-wicket partnership, making a mockery of England's protracted batting display of taking 173 overs to score their 348.
The collapse begins
However with the score 99 for 1, How was brilliantly caught for 39 by Matthew Hoggard in the deep off Sidebottom to spark the collapse.
The left-armer then had Fleming caught behind point by a diving Alastair Cook for 66 on the final ball of his ninth over, and then had Sinclair caught by a diving Cook for two on the first ball of his next over, before he trapped Jacob Oram in front for a duck to complete the hat-trick.
Ross Taylor was then caught and bowled for six by Panesar, who then trapped Kyle Mills in front for 11, before Patel and Vettori guided their side through to the close.
"I think it's great," said Fleming of Sidebottom and Panesar's bowling late in the day.
"It has breathed some life into a game that was meandering and I think it's given us maybe our best chance of winning.
"While it was spectacular to lose the wickets and you'd love to be in a position where you can dictate a little bit more, if you look a the overall picture of the game, it's our best chance of winning."