The visit of the German club to St Mary’s marked the culmination of pre-season and provided Ronald Koeman’s men with their most rigorous examination of the summer.
Although Leverkusen emerged with a 1-0 win, it was Saints who arguably controlled more of the game and created the better of the opportunities.
Davis said the experience, against opponents who are one step away from qualifying for the Champions League group stages, was an ideal one ahead of Sunday’s trip to Anfield.
“We’ve gradually built up in each and every game,” said the 29-year-old, who was one of Saints’ star performers last season.
“It was a good test against good opposition, and that will prepare us well for next week.
“But we’ve still got things to improve on, as you could see, and we’ve got a week to do that ahead of Liverpool.”
Davis added: “It’s just really getting that extra 15, 20 per cent maybe in the final third from us, that extra little bit of sharpness.
“We’ve still got a week to prepare for Liverpool, but I think, especially in the last 15 or 20 minutes, some space started to open up a little bit, we got in good areas, but maybe just that final pass, that final decision, was just a bit short in and around the box.
“Hopefully we look forward to another successful season.”