And the points were sealed shortly before the break when Salah tapped in after Shaqiri's free-kick hit the woodwork, although Virgil van Dijk's injury after the break marred the win for Klopp.
Gillespie believes that Manchester United are playing catch-up, which they will not have wanted, but he is unconcerned about suggestions they might miss out on a top four finish.
For the previous games against West Ham, Leicester and Paris Saint-Germain, Shaqiri had to settle for a bit-part role which consisted of a combined 32 minutes.
Defensive frailty was a major criticism of Liverpool last season, but the acquisition of Virgin van Dijk has helped steady the ship.
It was a routine victory in the end, with Southampton offering little by way of resistance to a Liverpool side who march on towards a testing set of fixtures against Chelsea.
45+3' - GOAL! Liverpool 1-0 Southampton (Salah).
Liverpool meanwhile had plenty of chances to increase their lead - sub James Milner had two very good shots in quick succession before Salah thought he had his second of the afternoon, only to be flagged for offside.
Liverpool: Alisson, Robertson, van Dijk, Matip, Alexander-Arnold, Henderson (C), Wijnaldum, Mane, Salah, Shaqiri and Firmino.
He made an instant impact with a spectacular overhead kick against Manchester United in MI but despite impressing in training, Shaqiri has been limited to just 27 minutes of Premier League action in four matches, plus a five-minute cameo in the Champions League.
Silva was on target within minutes, looping a header in, and City wrapped it up with a superb goal, Gundogan and Sterling linking and the former bending a brilliant finish into the corner. There were 11 after six league games...