Saints are third in the standings after their best-ever start to a top-flight season, even trumping the 1983/84 campaign in which they finished runners-up.
Mauricio Pochettino's side boast the best defensive record in the Barclays Premier League and are unbeaten in eight matches, although those records will come under pressure after the international break.
Southampton return to action in a top-of-the-table clash at leaders Arsenal, before returning to London to take on Chelsea the following weekend - matches that Shaw knows will prove Saints' mettle.
"The next two games we have got coming up will show whereabouts we are and whether we can challenge for the top," the highly-rated left-back said.
"If you take points from there, you know you are doing well.
"It is two of the biggest clubs in the league and hopefully we are going to go there positive to pick up points."
Collecting points in the capital would be an impressive feat, but Shaw's immediate attention is on representing his country.
The 18-year-old is currently on duty with the England Under-21s and put in an outstanding performance on Thursday evening as Gareth Southgate's side beat Finland 3-0 in Milton Keynes.
"It was very dominant," Shaw said of the manner of victory.
"We knew from when we played them out in Finland that it was going to be a hard game.
"But we knew when we play at our ground that we take full control and I think we did that today. Obviously in the first half I was told to stay back a bit because sometimes when we lost the ball my space was free.
"But in the second half they went down to 10 men and it allowed me to get forward more."
That did not mean Shaw was not bombing down the left in the first half, though - in fact, it was the left-back's venomous strike that saw Saido Berahino net his first goal of the night.
The full-back's effort was destined for the bottom corner until a fine reaction save, meaning his wait for a goal for club and country continues.
"Obviously we have got San Marino on Tuesday so you never know what might happen then," Shaw said.
"But I hope (my first goal) will be for Saints. I want to score at home."
- Sports & Recreation
- Barclays Premier League