Isak was making his third appearance for Liverpool since his British record £125 million move from Newcastle on transfer deadline day.
The 26-year-old had drawn a blank in his 58-minute debut appearance against Atletico Madrid and also failed to net after coming off the bench for the last 23 minutes of Saturday’s Merseyside derby win over Everton.
But Isak got off the mark in the 43nd minute as Southampton goalkeeper Alex McCarthy tried a short pass that was intercepted by Federico Chiesa, who picked out the Sweden striker to fire home.
It was a belated birthday present for Isak, who turned 26 on Sunday.
Isak, who had scored for Newcastle in their League Cup final win against Liverpool last season, was taken off at half-time as he builds up his fitness after missing pre-season training while his transfer was being negotiated.
Shea Charles equalised for the Championship side in the 76th minute with a close-range finish after a wayward header from Liverpool’s Wataru Endo.
Arne Slot had made 11 changes from the Everton game but substitute Hugo Ekitike saved Liverpool’s understudies from a penalty shoot-out when he finished off Chiesa’s pass in the 85th minute.
The French striker took off his shirt to celebrate and was dismissed for a second bookable offence.
Liverpool have made a perfect start to the season, with seven successive wins, including five in the Premier League to take them to the top of the table and one in the Champions League against Atletico.
Meanwhile, Chelsea were in danger of a humiliating exit after boss Enzo Maresca made eight changes following the 2-1 defeat at Manchester United on Saturday.
In their first meeting with Chelsea since 1911, Lincoln took the lead thanks to a Blues blunder in the 42nd minute.
Enzo Fernandez’s careless pass inside his own penalty area was intercepted by Ivan Varfolomeev before it reached the dawdling Malo Gusto.
Varfolomeev headed on to Rob Street and he guided a composed finish past Filip Jorgensen from 12 yards.
Chelsea had lost their last two games against United and Bayern Munich.
But Tyrique George got them back on track three minutes into the second half with a half-volley that crashed in off the post.
Facundo Buonanotte put Maresca’s men in front after 50 minutes, dancing through the Lincoln defence before slotting home.
Burnley crashed to a shock 2-1 defeat against third-tier Cardiff at Turf Moor.
Clarets boss Scott Parker paid the price for changing his entire team from last weekend’s game against Nottingham Forest.
Cardiff’s Joel Colwill prodded home in the 30th minute and Callum Robinson bagged the visitors’ second five minutes later.
Zian Flemming’s 56th-minute reply wasn’t enough to spare Burnley’s blushes.
Wolves lifted the gloom around Molineux with a 2-0 win against Everton.
Vitor Pereira’s team sit bottom of the Premier League after losing all five of their top-flight games.
Marshall Munetsi put Wolves ahead with a 29th-minute tap-in.
Toluwalase Arokodare chipped home Wolves’ second in the 87th minute as they avenged a 3-2 defeat against Everton in the Premier League in August.
Diego Gomez scored four times as Brighton thrashed third-division Barnsley 6-0 at Oakwell.
Emile Smith Rowe’s 66th-minute goal sent Fulham through with a 1-0 win against fourth-tier Cambridge at Craven Cottage. (AFP)
