In what way was petersburg a crucial victory for the union in the civil war?

Petersburg is really crucial for the victory because it's the main point for the Confederate's supply line.
By winning the war of Petersburg, the union managed to cut off food and ammo supplies for other soldiers in another battlefront which eventually caused the Confederate army to start falling like dominos.

The Siege of Petersburg sapped Lee's strength and led directly both to the loss of the confederate capital of Richmond and to Lee's surrender at Appomattox a short time after abandoning his defensive position. Petersburg was the site of the railhead supplying Richmond, so Lee was forced to defend it or abandon his capital. The siege was a significant strategic success because whereas Grant had access to practically unlimited resupply of material and troops, Lee's position only became weaker as the battle dragged on.

From mexican american war , us gained new mexico.