Victory over Montenegro at Wembley on Friday and another win at home to Poland next week will assure England of a place at the 2014 World Cup finals in Brazil.
Coach Holger Osieck defended his decision, saying he was "opening doors to talent" that caught his attention in a bid to test upcoming players at international level.
Schwarzer signed on a free transfer earlier this month after spells at Fulham and Middlesbrough and will provide valuable competition and motivation for number one goalkeeper Petr Cech.
Mourinho made clear that he regarded Cech as the world's best 'keeper, but he said the 40-year-old Schwarzer gave Chelsea extra "security" as a vastly experienced and highly rated back-up.
A reflective Jose Mourinho on Monday transformed himself into the "Happy One" as he embarked on his second spell in charge of Chelsea, outlining his hunger for fresh glory at the English Premier League side.
Cech believes the pressure will continue to mount until the very end of the campaign and Chelsea cannot afford any slip-ups against a Swansea side that beat them in the last four of the League cup earlier this season.
In the end, the only sour note for the visitors was a yellow card for Ashley Cole that rules him out of the second leg, although Basel will also be without centre-back Aleksandar Dragovic for the same reason.
Trailing 1-0 to Demba Ba's magnificent volley early in the second half, United looked poised to equalise in the 61st minute, only for Petr Cech to produce a stunning stop to repel Javier Hernandez's close-range header.
The Manchester United forward's first-half header wasn't enough to secure what would have been a valuable victory over the group leaders as Montenegro hit back to equalise through Dejan Damjanovic in the second half.
Montenegro coach Branko Brnovic has raised the stakes ahead of his side's World Cup qualifying showdown with England by claiming that Roy Hodgson's team are anxious about the game.