"The application throughout was excellent. Could we get between the lines? Those diagonal runs were a real threat," Southgate said after the match.
"There were some nice moments. Really pleased for Chilly, Jack's first goal, Tammy, Prowsey, Bukayo... They're all players who haven't scored that often for England, so it's a special moment for all of them."
Chelsea's Chilwell was forced into a long wait for his goal to be confirmed, but VAR ruled Jadon Sancho has been onside in the build-up to England's 17th-minute opener.
England's second was a sight that fans will be hoping to see for years to come as youngsters Phil Foden and Saka teamed up on 40 minutes.
With England passing the ball back and forth looking for a gap, Foden snapped the rhythm and sent Saka off down the right. He took a touch before hitting a scorching shot into the Andorra net.
Just before the hour mark, Sancho sent a low ball into the box which Abraham slipped into the net past Josep Gomes to add England's third.
England were awarded a penalty after Grealish was brought down in the 79th minute by Jesus Rubio. Ward-Prowse's intital shot was saved but he gathered the rebound and made sure to convert on his second attempt.
However, Southgate's side weren't finished, after being found by a long Sam Johnstone throw, Grealish displayed a moment of excellent individual skill to dink and jive around the Andorra defence and score on 86 minutes.
The match also historic as it marked the first time an England game was overseen by an entirely female refereeing team with Kateryna Monzul taking charge.