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.
In front of 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.
Djordje Despotovic and Aleksandar Pantic and England Under-21 debutant Raheem Sterling were all booked inside seven minutes and this 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 Tom 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.
For all of England's attacking intent, Serbia were becoming an increasing threat and a Luka Milunovic header at the back post was inches away from finding the top corner. At the other end, Jack Rodwell screwed wide from a free-kick and Ince saw a well-struck effort just curl wide, before Jordan Henderson came close with a 20-yard strike as half-time approached.
Loose passing and wayward finishing cost both sides in the second half, with neither goalkeeper tested until the 61st minute when goalkeeper Branimir Aleksic parried a fierce Wilfried Zaha drive.
Four minutes later and England had the lead after Adam 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 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.
- Raheem Sterling