Tipsarevic had a match point against the big-serving Raonic when leading 5-4 in the fifth set.
Djokovic, coming off a four-set loss to Rafael Nadal in the US Open final on Monday, won 12 consecutive games midway through his match and had no problem dispatching the 41st-ranked Canadian.
He said he was still recovering from the change of the time zones and jet leg after arriving in the Serbian capital from New York on Wednesday.
"I'm glad the match was short," Djokovic said. "I forced him to make mistakes. ... I find additional inspiration while playing for my country."
Pospisil said Djokovic played "an amazing" match and gave him no chance.
"There is no excuse," Pospisil said. "He is the No. 1 player in the world for a reason."
Canada has never made it this far in the Davis Cup, while Serbia won the event in 2010. Doubles will be played Saturday and reverse singles on Sunday.
The winner of the best-of-five series will face either the Czech Republic or Argentina in the final in November. The defending champion Czechs took a 2-0 lead against Argentina on Friday.
Czech Republic 2, Argentina 0
The Czechs are one win from their third final in five years.
Stepanek overcame a sluggish start in the first match, defeating Juan Monaco 7-6 (3), 6-3, 6-2. Berdych, ranked fifth, followed with a 6-4, 4-6, 6-3, 6-4 victory over Leonardo Mayer on the fast indoor hard court at O2 Arena.
Argentina is without star player Juan Martin del Potro, who skipped the Davis Cup this season.
"We are 2-0 up, that's what we wanted. Now, the most important thing is to get ready for the doubles," Berdych said.
Berdych looked relieved after he edged his 93rd-ranked opponent in more than three hours. He had to save a break point before converting his second match point when Mayer hit a forehand wide.
"I'm grateful for the result," he said. "The first two sets were not ideal from me."
Said Mayer: "I tried to attack as much as I could. He was better."
Stepanek was playing his first Davis Cup singles match this year just days after winning his second Grand Slam doubles title, at the US Open on Sunday.
"I was trying to be active," said Stepanek, who had 13 aces. "It was an important match for me. I faced a great opponent, but I wanted to use to my favor the surface and the fans."
Stepanek missed the first-round victory at Switzerland because of a neck operation and was able to play only doubles in the 3-1 quarterfinal win at Kazakhstan. He and Berdych played all five matches in last year's final in Prague and beat Spain 3-2.
Information from The Associated Press was used in this report.