Chelsea conceded a league-high 12 goals before the international break, but bounced back with a 5-1 victory over Bournemouth as the Blues climbed to 6th in the Premier League.
Alonso's side are not competing in European football this season after a poor finish in the last campaign. However, this only helps their chances of picking up the FA Cup or EFL Cup alongside fighting for the title.
Alonso led Bayer Leverkusen to an unbeaten Bundesliga title and the German Cup in 2023/24 but he is yet to replicate such success with either Real Madrid or Chelsea since.
The Bournemouth result showed Chelsea's attacking potential, but Alonso's side must become consistent as the West London side aim for silverware.
Speaking to Casinolyze, Heskey revealed that the first trophy will help Alonso ease the pressure on him, pressure which has crushed so many Chelsea managers before him.
"There's an expectation when you're at the bigger clubs. In the first year, I'd say you're bedding in, but you still need a cup. In the second year you challenge, and in the third year you have to win, it’s the same with Xabi Alonso.
"Winning things eases the pressure a lot. Securing the League Cup or the FA Cup early on eases the pressure and the burden on you.
"When I was playing, I went to three cup finals at Leicester, not many teams played their strongest eleven in the League Cup, and you could win it by March. It's an early trophy you can take pressure off on.
"With Xabi and the way results are going, pressure is always going to be there, that first trophy eases pressure, it eases the burden on. The League Cup could be the start of a successful period there."
Chelsea's most realistic route to a trophy is through the FA Cup or Carabao Cup, two competitions the club remain in, with the next clash coming against Liverpool later this month.