Liverpool right-back Alexander-Arnold limped off late in the second half after injuring himself while clearing the ball.
The 22-year-old looked emotional as he needed assistance to hobble around the Riverside Stadium pitch.
Alexander-Arnold, who finished the season in fine form, is one of four right-backs in the England squad after being selected despite reports he would be left out.
Kieran Trippier, another of the right-backs, was used at left-back against Austria, while Reece James and Kyle Walker were rested.
Unable to call on players from Chelsea, Manchester City and Manchester United following their European finals, Southgate sent out an experimental team.
It was Saka who staked a claim for a place in the starting line-up for their tournament opener against Croatia at Wembley on June 13.
The 19-year-old Arsenal winger was a lively presence on the left flank and capped his eye-catching display with a composed second half finish.
At the age of 17 years and 338 days, Jude Bellingham became the youngest player to start for England since Wayne Rooney in 2003, when he appeared against Liechtenstein aged 17 years and 321 days.
Bellingham was making his first England start after an impressive debut season with Borussia Dortmund, but the midfielder struggled to make his presence felt in an anonymous display.
England complete their pre-tournament preparations with a friendly against Romania back in Middlesbrough on Sunday.
Aside from his selection issues, Southgate will be alarmed that a section of the 8,000 crowd loudly booed the England players when they took the knee in solidarity with the Black Lives Matter movement before kick-off.
There was applause after the jeering but Southgate will hope the negative reaction doesn't become a distraction if it lingers during the tournament.
Alexander-Arnold wasted little time showing his class as he surged onto Harry Kane's raking pass and forced Daniel Bachmann to tip over his fierce drive.
Southgate will need Tottenham striker Kane firing on all cylinders if England are to fulfil their manager's target of reaching at least the semi-finals.
The England captain, who finished as this season's Premier League top scorer, wasted a good chance to break the deadlock when he ran onto Grealish's pass, but took too long to shoot and drilled straight at Bachmann.
England lacked a cutting edge in the first half but they carved open the Austrians to take the lead in the 57th minute.
Kane's pass picked out Jesse Lingard just inside the Austria area and his attempt to find Jack Grealish was diverted to Saka, who had the simple task of slotting into the empty net from close-range.
Like Lingard, Ben Godfrey was one of six players left out of England's final 26-man squad when it was announced on Tuesday.
But Godfrey still came on for his debut and was immediately rescued by his Everton team-mate Jordan Pickford when the England keeper scrambled to kick the defender's woeful backpass off the line.
Southgate had challenged Lingard to "show me I'm wrong", but the Manchester United forward was unable to replicate the fine form he showed on loan at West Ham.
England were grateful to Pickford for preserving their lead when he made a superb save to push Marcel Sabitzer's strike onto the bar.