The Lions were awarded a penalty try, which resulted in Brown receiving his 10-minute sojourn on the sidelines, then Nowell and Jared Payne both touched down while he was off the field as the visitors' sustained pressure finally paid off.
Flyhalf Dan Biggar added three conversions and two penalties in the Hamilton encounter for the Lions, who now head to Auckland to face New Zealand in the first match of a three-test series on June 24.
Captain Stephen Donald slotted two first half penalties for a Chiefs side put under constant pressure from the visitors, who retained the ball for long periods and smothered any attack.
"We're very happy, this was a tough fixture," Lions captain Rory Best said. "We came here tonight to mix it up. We felt we had a good pack, a good scrum and that we could maul them.
"But at the same time we didn't want to limit ourselves and when we get decent ball you see what the backline can do."
Lions head coach Warren Gatland had said beforehand that places for the test series were still up for grabs. However, few pundits believe that many if any of the side that eased past the Chiefs would feature in Saturday's clash against the All Blacks.
Ireland number eight CJ Stander was one of those who may have put himself in the frame for selection at the weekend as the Lions forwards controlled the game by building phases and forcing errors.
On the downside, they did lack any real creativity or game-breaking penetration in the backs, although Nowell's second try was arguably the best of their tour so far.
The Exeter winger finished off a sweeping 80-metre move following another lineout error from the Chiefs, when the backs committed defenders and swept up the field before Nowell cut back past four Chiefs players to stroll over.
The visitors also had Joe Marler yellow carded early in the first half for a dangerous shoulder charge, but they slowed the tempo of the game, forced the Chiefs into errors and held the ball for long periods.
The Lions had held a 13-6 lead at halftime but once referee Jerome Garces issued Brown the yellow card for collapsing a lineout drive and awarded a penalty try, the visitors put the game beyond doubt.
"They suffocated us and scored some great tries," Donald said. "Disappointing in front of a great crowd that we just couldn't get going but we have got no complaints.
"They were just too good."
The Lions suffered defeats to New Zealand Super Rugby sides Auckland Blues and Otago Highlanders earlier in the tour, but Tuesday's win and Saturday's 32-10 triumph over the Maori All Blacks will give them hope of competing in the test matches.
(Reporting by Greg Stutchbury; Editing by John O'Brien)
