The pair beat Jack Sock and Denis Shapovalov 6-3 7-5 at Palexpo, Geneva, as both teams aim to reach the 13 points that will secure victory on Sunday.
Despite finally winning an opening-day singles match at the eighth attempt, thanks to Sock upsetting world No.11 Fabio Fognini, Team World could not improve on previous years.
Singles victories for Dominic Thiem and Stefanos Tsitsipas had given the home side a 2-1 lead heading into the doubles, and Federer and Zverev ensured they would take a two-point advantage into day two.
I had two great coaches, one on the court and one of the sideline telling me what to do every point – Alexander Zverev
The result also saw Federer move further ahead in the all-time Laver Cup standings with 13 points to Sock’s 11.
“All of us tennis players travel around the world for 10, 11 months of the year,” the Swiss told his home crowd.
“There’s no place like playing at home, so this is a very special night for me.”
Sock had won 10 of his points in doubles over the last two years but failed to add to that tally with new partner Shapovalov, despite a host of chances.
The Team World pairing converted just one of 16 break points that came their way, including six set points in a single game in the second set.
Europe might have been vulnerable on serve but they were the more clinical, a lone break giving them the first set and a fizzing forehand return from Zverev edging them ahead early in the second.
They withstood heavy pressure, Federer picking a superb volley off his shoelaces as they denied World a break point for the ninth time, but a blistering Shapovalov return levelled things at 4-4.
The momentum was now well and truly with Team World and they pushed the Zverev serve to the limit at 5-4, only for the German to dig in and fend off six set points.
It was a huge blow to those in red, and almost inevitably they gave up the crucial break in the next game with a Shapovalov double fault.
Federer was not about to let the chance pass him by and he served out for a memorable win, punching his fist in celebration and embracing Zverev.
“I had two great coaches, one on the court and one of the sideline telling me what to do every point,” said Zverev, who was helped by both Federer and the watching Rafael Nadal.
“It was perfect. I could shut down my brain a little bit and do what they told me.
“They’re the two greatest of all time and I’m going to listen to them.”