The Reds edged two points ahead of title rivals Manchester City with a last-minute 2-1 win over Tottenham on Sunday, and Gillespie was full of praise for his old team's ability to come up with the result.
Jurgen Klopp's side had been far from their best - only having 48 per cent of possession - but found the net in the 90th minute via a Toby Alderweireld own goal after Lucas Moura had cancelled out Roberto Firmino's first-half strike.
"[Winning when playing badly] is the mark of a championship-winning side," Gillespie told Love Sport Radio.
"How many times did we see Manchester United do that in their pomp?
"At times, you look at these sorts of scenarios and think 'yes, they didn't play very well and yes they were exceptionally lucky at the end'.
"But that's what it takes sometimes to win the Premier League. Right throughout the season, you're going to have a balance.
"Sometimes you're going to play well and not get the result, but sometimes you're going to play poorly and get that little bit of luck."
Liverpool have the best defensive record in the Premier League this term, with Virgil van Dijk receiving widespread praise for his role in shoring up the back line since arriving in January last year.
And in attack, Sadio Mane, Firmino and Mohamed Salah have often decimated teams, but Gillespie thinks Klopp should aim to reinforce his midfield this summer.
"Liverpool have got these sorts of players but Mane was a bit quiet yesterday," the former Reds star continued.
"You had flashes but he has been the one inspiration this season that has taken up the mantle.
"There's always that little bit of brilliance within him or the likes of Firmino, just that one turn or little pass which can be the difference. We have got players there.
"I would just like to see them have a couple more players in the middle of the park with that sort of ability to change a game.
"I'm sure Jurgen has identified that and come the summer, that might be where you strengthen, that area of the pitch."