Midfielder James Harper scored the 88th-minute winner from 25 yards as Argyle scrambled away Jamie Ward's corner from the left.
Harper had forced the corner with a stinging strike that home goalkeeper Romain Larrieu was forced to acrobatically push around the post.
Delighted United boss Blackwell said after the 1-0 win: "We knew it was going to be a tough game. We had to be resilient, strong and focused and I am delighted that we showed those qualities really.
"The defence was superb. The conditions out there were not conducive to good football, it was raining and the wind was howling all over the place.
"It makes it difficult. That's why teams don't like coming down here because you have to keep going and stay focused. It's important.
"We were putting them under good pressure towards the end and in the end we deserved to nick it but I would not have been unhappy with the draw. It was a great strike by James."
Plymouth manager Paul Sturrock was left to rue what might have been for his team.
Just four minutes before Harper had his decisive say, home captain Carl Fletcher had powered a similar effort to the eventual winner from the edge of the Sheffield United penalty area but the ball struck the post.
Sturrock said: "Football can be cruel sometimes. We lost to a stoppage-time goal at Leicester and we lost to a goal near the end again today.
"Carl has hit the post with a shot and then they go up the other end and score the winner. That's football, especially when you are down in the table where we are at the moment.
"What did please me though was that I was able to field the back four that had been miserly before and I did think there would only be a goal in the game today.
"If we had got that goal I think we would have gone on to win the game but we didn't and they did."
The best chances in a dour, weather-spoiled encounter came in the second half.
Argyle's on-loan Aston Villa central defender Shane Lowry headed just over from Alan Judge's right-wing corner after 47 minutes.
United countered with midfielder Nick Montgomery sending a thumping shot off the foot of the post from 25 yards 10 minutes later.
Plymouth playmaker Judge then bounced a chip off the top of Carl Ikeme's net, resulting in the goalkeeper suffering a hamstring injury as he stretched for the shot.