The package, which includes measures such as restricting family reunions for some migrants, was approved with a large majority of 429 votes to 147 against.
The new rules also lower the hurdles for the expulsion of convicted foreigners - a key measure proposed after the New Year's rampage in Cologne, where hundreds of women reported being sexually assaulted and robbed in a crowd of mostly migrant men.
Chancellor Angela Merkel has come under intense pressure to reduce the number of asylum seekers in Germany, which saw an influx of around 1.1 million in 2015.
Under the new rules, some migrants will be blocked from bringing their families to join them in Germany for two years.
The measure applies to some rejected asylum seekers who still cannot be expelled because they risk torture or the death penalty in their own country.
The asylum application process would also be sped up, with centres set up to process applications from "economic migrants" who have little prospect of winning refugee status.
Berlin's aim in gathering these applicants in specific centres is to facilitate their quick repatriation, to free up resources for war refugees who urgently need protection.