An injury to Shaw at the start of the second half led to a lengthy delay, with the Manchester United fullback withdrawn on a stretcher following a heavy collision with Carvajal, to be replaced by Danny Rose.
With many fearing the worst for Luke Shaw, it was reported shortly afterward that he was doing well, and had been contacting his family from within the England dressing-room to inform them of his condition.
"We hung in there and in the last 25 or 30 minutes we really came into the game, kept the ball a lot better and created some good chances and could well have got a draw out of it".
Manchester United forward Marcus Rashford opened scoring for the Three Lions but a goal each from Saul Niguez and Rodrigo ensured the 2010 World Cup winners emerge victorious.
Southgate said he was unsure whether his side could close the gap to the top teams before the next European Championship, in which the semi-finals and final will be played at Wembley.
A marauding run from Shaw down the left was finished off by Rashford at the back post thanks a to a delicious pass in yet another signal of their combined quality.
"We need to asses over the weekend, but I think it's likely we'll need to call some players in", Southgate said. We both have the same objective and before judging someone I like to meet them for myself in person.
Captain Sergio Ramos added: "We have started a new era with a coach who has a lot of experience and there are few better ways to start than to win at Wembley".
"I'm a fighter, I'll be back soon", he wrote on social networks after the game in an attempt to reassure fans.
Harry Kane insists England still have plenty of work to do to progress further and compete regularly with the world's elite as they prepare for their first game since a memorable World Cup summer.
However, a clash with Dani Carvajal early in the second half left him apparently unconscious on the turf.