Reid opened the scoring in the fourth minute and then looped the ball forward for Lewis Alessandra to cleverly lob City keeper Paul Farman a minute later, after Argyle keeper Jake Cole made a crucial save to deny Tom Miller.
Reid converted a 23rd minute penalty following Miller's foul on Tope Obadeyi and then completed his treble in the 35th minute with another measured finish.
Alessandra's 64th minute goal-bound effort was helped over the line by Imps defender Andrew Boyce as Argyle teed up a home tie against Conference side Welling United over the weekend of December 7/8.
"I thought it was a very good, very professional performance," he said. "Obviously the main thing was getting through the tie but I thought some of the stuff we played was outstanding.
"I don't take into account that Lincoln are in the Conference, a league below us. The way we passed the ball was very positive and to get five goals with a clean sheet can only give us a bit of confidence.
"I thought we had the game won early in the first half, and my two strikers did well with Reuben getting three and Lewis two.
"A hat-trick is a hat-trick no matter who you score against, it's in the record books and hopefully it will give Reuben a big lift. I thought he played really well, especially in the first half, he was real threat.
"At 1-0 Jake has made a great save and without that it could have been different but I think we totally dominated the game. We were four-up at half time but I said to them then that I would like them to go on and get more goals.
"In most games when you get four or five you sometimes take your foot off and get a bit lackadaisical. We did at times but overall it was a very good performance."