The match was due to get underway at 2:30 pm local time (1330 GMT).
However, with rain falling for several hours in Manchester on Sunday morning both the toss and match itself were put on hold, with deep puddles forming on the covers protecting the pitch and square.
The official cut-off time for a five-overs per side match, the minimum length required to force a decisive result, was 5:46 pm (1646 GMT).
But a fresh downpour shortly before a planned pitch inspection led the umpires to abandon the match at 4:18 pm (1518 GMT).
The three-match series finished level at 1-1 after Liam Livingstone's 87 helped England win the second T20 in Cardiff on Friday following Australia's commanding 28-run success in Wednesday's opener in Southampton.
England now face 50-over world champions Australia in a five-match one-day international series starting at Nottingham's Trent Bridge on Thursday.
