The Blues dropped two points as they were held to a goalless draw against a relegation-threatened Saints side who kept a clean sheet for the first time in 11 Premier League games.
Goalkeeper Angus Gunn, making his Premier League debut, twice denied Eden Hazard and Alvaro Morata had a strike ruled out for offside while also missing other opportunities as Sarri finally appeared to lose patience with the misfiring striker at Stamford Bridge.
He said: "On the bench I tried to do my job. I think we played a very good match for 80 metres of the pitch, then we were in trouble in the last 20 metres. We have to try to solve the problem of the last 15, 20 metres."
Sarri revealed Olivier Giroud's ankle injury is not as bad as first feared, but he dismissed the prospect of recalling Michy Batshuayi from Valencia or Tammy Abraham from Aston Villa.
The Italian suggested his preference was to recruit a new striker, but given Chelsea's board, chiefly director Marina Granovskaia, are responsible for transfers, it is by no means certain Sarri's wish will be granted.
He added: "The club knows very well my opinion. I think we need something different for characteristics. I'm not in charge of the market.
"I have to try to improve my players, my team, my offensive phase, or better my offensive phase in the last 20 metres.
"The club knows my position, my opinion. It's up to the board."