Argentine icon Messi, 38, opened the scoring in the 10th minute, added another goal in the 39th and assisted on compatriot Allende's goals in the 73rd and 76th that sealed Miami's triumph.
Inter Miami won the best-of-three Eastern Conference first-round series 2-1 to book a second-round match against FC Cincinnati, who defeated Columbus 2-1 in another series-deciding encounter.
Eight-time Ballon d'Or winner Messi ensured Miami would avoid a third first-round exit in four seasons after having fallen to Atlanta last year and New York City in 2022.
In the night's other match, 10-man Minnesota United rallied to eliminate Seattle 7-6 on penalty kicks after playing to a 3-3 draw in a Western Conference thriller, booking a second round match against Sunday's winner between Portland and top seed San Diego.
Inter Miami was without star Uruguayan striker Luis Suarez, who was suspended one game by MLS for violent conduct in last week's loss at Nashville.
MLS season goal-scoring leader Messi lifted Miami ahead early, taking a deflected ball and racing to the top of the box to fire a left-footed shot into the lower right corner.
Messi doubled the margin six minutes before halftime when he fired home a left-footed shot from just outside the box off a pass from 19-year-old Argentine rookie striker Mateo Silvetti.
Allende fired in his first goal from point-blank range off passes from Messi and Jordi Alba then took a long pass from Messi and chipped a high ball into the net.
Just a couple of minutes later, Allenda would find the back of the net for his second goal of the game, adding some extra insurance en route to Miami's win over Nashville.