The Italian duo, seeded number two for the tournament, brushed aside Kristina Mladenovic and Timea Babos 6-1 6-3 on Centre Court, collapsing to the turf in victory after becoming only the fifth pair in history to achieve the feat.
After triumphing at Roland Garros and the US Open in 2012, Errani and Vinci claimed a third grand slam with Australian Open success last year.
They defended that title in January, but still required the Wimbledon crown to complete the set and the duo did just that on Saturday, crushing their 14th-seed opponents with a minimum of fuss.
The Italian pair came from a set down in their first-round match against Lyudmyla and Nadiya Kichenok, but there were no such scares on Saturday in what was a clinical performance.
Errani and Vinci broke Mladenovic and Babos in every game in the first set, and claimed the crucial breakthrough in game six of the second before going on to see out victory, becoming the first pair to seal the career grand slam since the Williams sisters in 2001.