The 23-year-old winger was introduced in the 66th minute of the match at Wembley and immediately caused San Marino problems with his pace and trickery before getting on the scoresheet with 18 minutes left.
Collecting the ball on the right, Andros shifted it onto his left foot and fired inside the near post from the right-hand corner of the 18-yard box.
The goal came two days shy of a calendar year since his first England strike – a long-range stunner on his Three Lions debut against Montenegro in a World Cup qualifier last October.
England had taken a 2-0 lead in at half-time against San Marino, courtesy of Phil Jagielka’s header and a Wayne Rooney penalty. Danny Welbeck made it 3-0 four minutes after the break before making way for Andros, while a cross from Rooney went in off Alessandro Della Valle to complete the scoring.
Andros might have added a sixth goal two minutes from time, but his fierce low shot from the edge of the box was blocked on the line by defender Cristian Brolli.
England face Estonia in a further Euro 2016 qualifier on Sunday, kick-off 5pm.
Elsewhere, young defender Dominic Ball helped England Under-20s get off to a winning start to the Four Nation Tournament in the Netherlands with a 1-0 win against Germany on Thursday afternoon. Dominic played the full game, helping his side keep a clean sheet.
Meanwhile, Kenny McEvoy went on as a half-time substitute for Republic of Ireland Under-21s in their 4-1 friendly defeat to Norway.