But Sam Allardyce's team were taken the distance by a resilient Leicester at the Reebok Stadium.
With a minute remaining of the first period of extra-time after both sides had failed to score, Borgetti fortuitously found the back of the net.
Ivan Campo's cross was headed back into the six-yard box by Radhi Jaidi at the back post but Dion Dublin's clearance hit Borgetti and flew into the net.
Vaz Te doubled Bolton's lead a minute into the second period of extra-time, holding off two Leicester defenders to toe-poke home.
However, Leicester didn't lie down and Gareth Williams beat Bolton keeper Jussi Jaaskelainen with a rasping left-foot strike nearly 30 yards from goal.
Jaaskelainen also made a point-blank save in the last minute of extra time, but thankfully for Bolton they had done enough to progress.
Bolton were far the better team in the first half and the home side nearly took the lead with their first clear-cut chance in the eighth minute.
Midfielder Jay-Jay Okocha shot over from the edge of the box after a clever lay-off from Martin Djetou.
Two minutes later Vaz Te, who proved a handful for Leicester in the opening exchanges, had a sliding shot at the back post but defender Nils-Eric Johansson was on hand to avert any danger as the ball rolled invitingly across the six-yard box.
Japan international Hidetoshi Nakata provided a deft pass to Vaz Te moments later, but his left-foot drive was brilliantly saved by Leicester keeper Rab Douglas who denied him with his legs.
A minute later Wanderers skipper Gary Speed had a shot charged down on the edge of the area by centre back Johansson and as Bolton pressed, Leicester looked ever more vulnerable.
Moments later Nakata's fierce shot was deflected over the bar from Dublin.
Three minutes later Nakata scuffed a shot over the bar, although he did well to connect when a high bouncing ball that fell in his path on the half-volley.
Vaz Te had another chance to score but he fired well wide from the edge of the area.
Leicester's first chance of the half came in the 21st minute.
Former Manchester City utility player Danny Tiatto crossed from the left but Mark De Vries headed wide at the back post.
Bolton continued to dominate possession and in the 32nd minute Okocha fired well over from distance as the home side looked to break the deadlock.
Tal Ben-Haim headed over two minutes into the second half after Nakata's corner from the right, before Vaz Te fired wide with a left-foot shot on the hour mark after Kevin Davies' flick on.
Davies' delicate flick with the outside of his right boot found Ricardo Gardner in the 70th minute but he was deemed offside when he missed the target from close range.
Djetou's cross from the left was taken down by Vaz Te, who injured himself in the process, but substitute Campo, back after a broken foot, somehow missed from the edge of the six-yard box.
In the 82nd minute Nakata had a shot blocked by Johannes Gudjonsson and Vaz Te pushed the rebound wide as Bolton's frustrating night continued.
In injury time Leicester substitute Momo Sylla fired over from inside the penalty area but that was the way it stayed as the match went into extra-time.