Gylfi Sigurdsson had put Reading ahead with a tap-in before Lloyd Dyer levelled with a long-range strike.
In midfield Gylfi Sigurdsson came on leaps and bounds for Reading and scored lots of goals too.
Gardner also had a shot saved by keeper Gunnar Sigurdsson and an effort deflected wide from 10 yards.
Vertonghen becomes new Tottenham manager Andre Villas-Boas' second signing after Icelandic midfielder Gylfi Sigurdsson, who joined from Hoffenheim last week.
Tottenham joined it on 58 by snatching a late 2-2 draw at home against Everton, thanks to Gylfi Sigurdsson's 87th-minute equalizer.
Kristjan Orn Sigurdsson pulled a goal back for Iceland with two minutes remaining, but Bert van Marwijk's team comfortably secured the victory.
The 21-year-old had a fierce drive saved by Sigurdsson and was stopped from connecting with Marlon Harewood's low centre, before opening the scoring.
And he would be made to pay for that miss as Sigurdsson doubled his account with a fierce volley into the roof of the net following a corner.
Reading looked the most likely to make a breakthrough as Howard shot over the crossbar, Sigurdsson's looping header dropped just wide and Foxes keeper Conrad Logan saved from Kebe.
Sigurdsson, who plays in the English Premier League with Swansea, grabbed his first from the penalty spot after Stefan de Vrij fouled Birkir Bjarnason, firing powerfully past Jasper Cillessen.
Iceland midfielder Sigurdsson was unfortunate eight minutes into the second half when his free-kick from 30 yards out came back off the post with Jamie Ashdown beaten.
There were a few tense moments for Scotland immediately after Indridi Sigurdsson equalised for Iceland and in the closing stages, when the heroics of Gordon came to the rescue.
It was striker Harewood again, who sped up the right and cut the ball back for Gardner, who buried the ball high past the reach of Sigurdsson from 20 yards.
Sigurdsson then opened the scoring on 22 minutes, with the Icelander given the simple task of turning the ball in from close range after good work by Jimmy Kebe down the right wing.
An own goal from Ryan Bertrand had given Liverpool the lead but an injury-time penalty from Gylfi Sigurdsson and an extra-time header from Shane Long turned the game in Reading's favour.
Just like he was in the win over Latvia on Friday, Gylfi Sigurdsson was Iceland's hero, scoring twice in a famous win over a side that finished third at the 2014 World Cup in Brazil.
Gylfi Sigurdsson and Birkir Bjarnason opened up the visiting defence and Kolbeinn Sigtorsson fired in a shot from the near post which Martin saved before also blocking the follow-up from Sigurdsson on the penalty spot.
With just 10 minutes left, substitute Gylfi Sigurdsson leveled for the visitors after a cute Adebayor backheel but despite a late free-kick from Gareth Bale, Spurs had to settle for a draw that may ultimately seem disappointing.
Duff shot wide with a volley from the edge of the area before Tugay went close with an outrageous 40-yard shot but otherwise the West Brom back three of Darren Moore, Phil Gilchrist and Lars Sigurdsson looked comfortable.
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