Scott Pollock opened the scoring from the penalty spot, before a Jake Wright brace and a Will Atkinson goal sealed the Pilgrims’ place in the next round.
“I'm delighted for the group,” he said. “This is down to the boys and how they are performing.
“We are giving them plenty of details, but they are taking it all in and being brave on the ball.
“We look like a team who are starting to control games. They asked questions of us and had little spells but we came through that. I felt we were in control and that's no discredit to Lichfield.
“The group are recognising periods of the game and are scoring goals at the right times.
“Jake (Wright) has been out for a while but you can see what he brings to the team, he has so much enthusiasm.
“I’m pleased for him, it will give him the world of confidence.”
United have no midweek game this week with Culverhourse delighted to be able to do extra work on the training ground.
“We will get two training sessions in before going to Telford,” he added. “We will give them more detail. They are receptive at the moment and long may it continue.
“We are getting players back and have competition for places, that is what we need. We can’t let people be comfortable, they have to be under pressure to keep their shirt.
“Our crowd was superb today, this is a fantastic club and we will try to give them a team they deserve.”
Wright was named in the United starting XI - at the expense of Byron Harrison - as Ian Culverhouse made one change to the side which had beaten Buxton 3-0 on Tuesday evening.
United started brightly at the Trade Tyre Community Stadium and took a 16th minute lead when Wright was flattened by keeper James Beeson and Pollock sent the Lichfield custodian the wrong way from the spot for his fifth goal of the season.
Joe Leesley went close with two first half free-kicks - the first rippling the roof of the net and the second being blocked by the Lichfield defensive wall.
Boston took a major step towards the next round with a goal of real quality six minutes before the break. Atkinson and Zak Mills were involved heavily, with Keenan Ferguson delivering a fine ball from the right - and striker Wright buried a brilliant diving header from close-range.
Sam Long was called upon five minutes after the break to deny Luke Childs from distance but United always looked likely to add to their tally and they did that twice within a five-minute spell midway through the half.
Alex Brown delivered a ball of real quality from the left and Wright dispatched another fine headed effort from 10 yards.
Five minutes later, Jordon Crawford got free on the right-hand side to put the ball back for Atkinson to roll home his first goal in United colours.
Burnside was dismissed following two yellow card fouls on Brown and although Luke Shiels had a late header ruled out for a push, United were long since safely through.
Lichfield (4-4-2): James Beeson; Sam Walton (Kyren Hamilton 62), Jamie Elkes, Joe Haines, Sam Goodby; Luke Childs, Kyle Baxter (Harrison James 86), Cameron Dunn (Luke Keen 68), Lewi Burnside; Ethan Muckley, Sam Fitzgerald. Subs (not used): Harry Crook, Greg Lewis (gk).
Boston (4-1-2-1-2): Sam Long; Keenan Ferguson (Ellis Myles 55), Luke Shiels, Ben Pollock (Jean Leroy Belehouan 59), Alex Brown; Joe Leesley; Zak Mills (Micah Obiero 69), Will Atkinson; Scott Pollock; Jake Wright, Jordon Crawford. Subs (not used): Brad Abbott, Byron Harrison, Rhys Davies (gk).
Referee: Darius Bradley.
PIC: Tim Hoff.