We quickly overcame the shock of going a goal behind in the fourth minute to dominate long periods of the game at White Hart Lane.
Dawson soon levelled and when Adebayor's fine header put us ahead early in the second half, it appeared that would be that, but we couldn't find the third goal and Hugo Lloris was forced into important saves from Pavel Pogrebnyak and Jimmy Kebe at 2-1.
Thankfully, Dempsey fired home the third via a deflection just three minutes after coming on as substitute, and a seventh win in nine matches in the Premier League was in the bag.
This third-successive victory after away success at Villa on Boxing Day and Sunderland on Saturday also moved us back into third.
Andre Villas-Boas made two changes from Sunderland with Jan Vertonghen back in at centre-back alongside Dawson and Gylfi Sigurdsson given the nod to replace the suspended Gareth Bale.
Buoyed by a win against West Ham on Saturday, Reading found themselves in front after four minutes.
Pogrebnyak was brought down in an inviting positon for Ian Harte's trusted left foot and the full-back delivered, his free-kick cannoning back off the crossbar and falling kindly for Pogrebnyak, who stooped to head home the rebound.
We slowly took control. Adebayor headed just wide from Aaron Lennon's cross and it was soon 1-1, as Dawson darted to the near post to glance home Sigurdsson's corner, 10 minutes on the clock.
The play took place in Reading's half for the rest of the first half, as shots rained in from all angles without further reward.
Jermain Defoe and Mousa Dembele both tried their luck and then Dembele fired just wide from 25 yards after holding off Jem Karacan.
Sandro was next up on 34 minutes, another effort that sailed just wide and Defoe went even closer after side-stepping Mikele Leigertwood and whipping a shot back across Adam Federici.
The visitors' goalkeeper was busy for the rest of the half, denying Defoe, Dembele and then his best save of the half, sharp reactions to his left after Sandro's 25-yarder clipped off Leigertwood.
We were soon on the attack again in the second half as Defoe and Federici locked horns again, this time the goalkeeper taking the striker's curler to his right.
There was little Federici could do about Adebayor's goal on 51 minutes, a textbook header. Lennon got to the right byline and chipped up to the far post where Adebayor, reaching back, powered his header home.
Sandro fired just wide again, Lennon had two efforts blocked, Defoe didn't connect properly with another chance and Kyle Naughton's volley fizzed past the post as we piled on the pressure.
What would have been a clinching third goal didn't arrive though and suddenly, Reading carved out chances to equalise.
First, Pogrebnyak bundled past Vertonghen, Lloris saved well from the angle but the ball bounced up back to the striker, who headed wide.
Next, Kebe burst through on goal onto Danny Guthrie's pass only to be denied by a combination of Lloris and Kyle Walker sliding in.
Alex Pearce then connected with Harte's corner, planting the ball back into the danger atea where Dawson did enough to divert it away from Pogrebnyak.
Dempsey entered the fray for Sigurdsson during this spell and he didn't have to wait long to play the winning hand.
Three minutes after arriving, the USA striker picked the ball up in space 25 yards out, let fly and looked on in delight as the ball struck Leigertwood's outstretched leg and looped up and over Federici.
The final action saw Federici excel again, first to block Defoe one-on-one and then his best save of the day, full stretch to tip over Scott Parker's 20-yarder.
Spurs (4-4-2): Lloris; Walker, Dawson, Vertonghen, Naughton; Lennon, Dembele (Parker, 70), Sandro, Sigurdsson (Dempsey, 76); Adebayor (Livermore, 88), Defoe. Unused subs: Friedel, Caulker, Falque, Huddlestone.
Reading (4-5-1): Federici; Gunter, Mariappa, Pearce, Harte; Kebe, Leigertwood, Karacan (Robson-Kanu, 57), Gutherie, McAnuff (Le Fondre, 73); Pogrebnyak (Hunt, 84). Unused subs: Taylor, Morrison, Tabb, Cummings.
Goals: Spurs - Dawson (10), Adebayor (51), Dempsey (79). Reading - Pogrebnyak (4).
Referee: Mr H Webb.