The 32-year-old returned figures of 5-27 on the third day, producing a spell that helped swing the World Test Championship contest in the hosts' favour in Trinidad and Tobago, before a late comeback from Pakistan.
Greaves surpassed the previous record of four consecutive wicket maidens, set by former England fast bowler Stuart Broad against South Africa in Johannesburg in 2016.
Playing in his 17th Test, the Barbadian claimed his maiden five-wicket haul as Pakistan were bowled out for 282 in reply to West Indies' first-innings total of 311.
"When I came on captain Roston Chase just said, 'Continue to be disciplined in what you are doing' and to see if I can get some wickets for the team," Greaves said.
"The ball nipped around and I got some reward. Anytime I get the ball in hand the team looks to me to be that solid player for them (and) try to dig us out of situations.
"To get five wickets for the first time in Test cricket, I'm really happy, but there's still a lot of work to do in the game."
Pakistan were in command at 3-244, with Shan Masood having reached a century, before Greaves triggered a dramatic collapse.
The seamer dismissed Masood for 109 then struck in successive overs to remove Aamer Jamal, Ali Usman and Mohammad Rizwan without conceding a run, leaving Pakistan struggling at 8-267 at lunch.
Greaves completed the historic feat after the break when he had Mohammad Abbas caught behind, giving him five wickets from five successive maiden overs. Pakistan were eventually bowled out for 282, trailing by 29 runs on the first innings.
Pakistan fought back late in the day, reducing West Indies to 7-126 in their second innings by stumps. The hosts nevertheless held an overall lead of 155 runs heading into the fourth day.
