Saints started the match much the better side and after nine minutes last weekend’s try-scorer Juarno Augstus ploughed through the middle of the pitch to tee up on-rushing second-row Coles to score the opener.
Player of the match Smith added the extras before two penalties meant Saints raced into a 13-point lead before the half-hour mark.
Chiefs fly-half Harvey Skinner then found a yard to dart through the Saints defence and found Woodburn, who had the simplest of tasks to score under the posts.
In the second half, Exeter’s stand-out player, Josh Hodge, found his range with the boot wonderfully.
His monster kick from just over halfway clipped the inside of the right upright to reduce the arrears to three points.
The visitors were then dealt a hammer blow when Munga charged down scrum-half Sam Maunder’s attempted box kick to crash over the whitewash on his 50th Premiership appearance.
Four minutes later, a pinpoint kick from the impressive George Furbank found Hutchinson on the left wing and he finished expertly to score the try of the match.
Smith, who ended six from six with the boot, then put Saints 20 points clear with a well-taken drop goal after the hour mark.
Feyi-Waboso then scored his first try with 11 minutes remaining after it was adjudged Woodburn had not deliberately knocked the ball on, following a lengthy TMO review which showed a fingertip touch from Saints replacement Tom West before it got to the Chiefs winger.
With four minutes left, Hutchinson was sent to the sin-bin for a high tackle on Woodburn and moments later substitute Tom Cairns did brilliantly to find Feyi-Waboso to score his second and reduce the deficit to just six.
The comeback proved to be too little too late for the Chiefs, however, as the hosts held on to claim a confidence-building victory.
