Liverpool landed the Italy forward on a four-year deal from Juventus earlier this week for an initial fee of £10million, possibly rising to £12.5m with add-ons.
The 26-year-old can play anywhere across the front three and Carragher expects to see him filling in for Mo Salah on the right on multiple occasions this season.
Former Reds captain Carragher was at Old Trafford on Sunday as part of the Sky Sports punditry team for Liverpool's 3-0 win at Manchester United.
Chiesa was not in the Liverpool squad for that game but he was in the crowd as a spectator.
Liverpool stars Federico Chiesa (left) and Mo Salah (right) pictured training together on Friday
Carragher was asked ahead of kick-off if he thought that Chiesa's arrival would 'add a little bit of pressure' onto Liverpool's other attacking players.
'I think it does a touch,' Carragher replied. 'I think the biggest thing it gives Liverpool is actual genuine quality to replace Mo Salah, when he is rested at times or maybe has a slight knock.
'Because when you think of all the other attacking options Liverpool have, the majority of them feel more comfortable on the left. So any of them when they play on the right - whether it's [Darwin] Nunez or [Diogo] Jota or [Luis] Diaz or [Cody] Gakpo - it never feels right.
'Chiesa, more often than not, is playing from that side, so I think it is a great addition.
'We talk about Manchester United in the past having four forwards [for two striking spots in 1998-99] famously in the treble season. Liverpool play with three forwards and they've got six now, so there's a healthy competition.'
Salah scored United's third goal on Sunday after a first-half brace from Luis Diaz.
He has now netted 10 goals in nine appearances at Old Trafford for Liverpool in all competitions.