The West Brom defender's fifth goal in nine appearances for the Young Lions proved the difference in a game Stuart Pearce's side dominated for the most part at Carrow Road.
England impressed during the first-half in Norwich but could not find a way to break the deadlock, with Tom Ince and Jordan Henderson coming closest.
Luka Milunovic's header was the best chance Serbia could produce in the opening 45 minutes and, after a poor start to the second half, the hosts' dominance paid in the 65th minute.
Right-back Adam Smith drew a foul in the box and Dawson stepped up to lash home the resulting penalty to the delight of most the 17,266 spectators.
Serbia threatened a later equaliser but could not find a way past England's defence, who held out for their fifth clean sheet in six games to give them the advantage heading into Tuesday's second leg in Krusevac.
It took just 38 seconds for the match to come to life, with lone striker Djordje Despotovic booked for a late lunging tackle that caught Dawson.
The challenge was an early warning sign of the Serbian's intent to heap pressure on the hosts.
It also highlighted the ease in which referee Deniz Aytekin would dish out cards, with Aleksandar Pantic and England Under-21 debutant Raheem Sterling both added to the German's notebook inside seven minutes.
The disruption to play meant it took a while for the game to flow, with the first chance of note coming 13 minutes in.
Sunderland loanee Danny Rose cut in from the left and floated in a wonderful cross, although the unmarked Ince could only direct his header wide.
It began a spell of concerted England pressure and Marvin Sordell's cross from the left would have been tapped home had it not been for captain Slobodan Medojevic's block.
Jack Rodwell tested Branimir Aleksic as the Young Lions continued to push for an opener, which would have come had the Serbia goalkeeper not rushed off his line to deny Ince's burst on goal.
For all of England's attacking intent, Serbia were becoming an increasing threat and a Milunovic header at the backpost was inches away from finding the top corner.
Sasa Markovic and Darko Lazovic were the next to get away attempts for the Eastern Europeans, who had Filip Mladenovic booked six minutes before the break for taking down Smith.
Rodwell screwed wide from the resulting free-kick and Ince saw a well-struck effort just curl wide, before Henderson came close with a 20-yard strike as half-time approached.
Jason Lowe replaced Rodwell when the teams reappeared for a second half that began sloppily.
Loose passing and wayward finishing cost both sides, with neither goalkeeper tested until the 61st minute when Aleksic parried a fierce Wilfried Zaha drive.
Four minutes later and England had the lead after Smith was caught just inside the penalty box by Milunovic.
Dawson stepped up to lash the resulting spot-kick into the bottom right-hand corner past the outstretched Aleksic.
England came close to doubling their advantage shortly afterwards when substitute Andros Townsend - on for his Under-21 debut in place of Sterling - fired a low cross to the backpost, where Wilfied Zaha picked the ball up and placed just wide.
Serbia searched for a leveller as the clock began to wound down, with Nemanja Gudelj coming close with a long-range free-kick before Medojevic was booked.
Milunovic fired over as the Eastern Europeans continued to threaten, but at the other end Aleksic produced a fine one-hand save to deny Steven Caulker.
With the match coming to an end, Dawson provided a superb late block to deny Despotovic and ensure victory for England and a 1-0 advantage.