Exeter had put the pressure on by beating London Irish to move into the play-off positions by the time play got underway at the Rec.
But if the Saints were feeling the heat they weren't showing it, putting on a superbly professional performance to control play from the off. In the first 20 minutes they denied Bath either possession or position, working their way into a 10-point lead that was rarely threatened.
Prompted by outstanding performances from half-backs Lee Dickson and Ryan Lamb the visitors mixed and matched their attacking play from the first whistle. The forwards pounded away at the fringes, the backs spreading play with pace and ambition.
Nevertheless it was at the set-piece where Saints got the upper hand immediately. The scrum has won the headlines this season but it was the lineout which put Bath on the back foot early on, James Craig stealing two balls against the throw to spoil home possession and get his team going forward.
With 12 minutes on the clock the scrum took over. A Lamb grubber forced Nick Abendanon to concede a five-metre scrum. The first two ended in penalties, and sensing blood Dickson opted for the same again both times.
The next two were re-set, but at the fifth time of asking Saints got the nudge on once and for all, referee Martin Fox's patience ran out and the penalty try was the result. Lamb's conversion was infinitely easier than it would otherwise have been - from in front of the sticks instead of wide on the right - and the visitors had a well deserved lead.
Having gone in front through the pack Saints then gave the backs a chance. This looked promising, too, Craig's lineout steal winning a penalty and giving Lee Dickson the opportunity to keep the tempo up.
Ben Foden put Chris Ashton away down the right wing and George Pisi popped up a couple of times on the left. But while Bath's defence held firm and won a penalty, their lineout was proving their Achilles' heel, Saints again winning the ball back, this time on their own 22.
Even when Bath managed to win their own ball they could not make the most of it, fumbling the ball in the maul. Indeed this even came close to giving Saints the impetus for a second try. Man of the match Foden recovered his own up and under superbly and Diggin was put in the clear down the left, but when the winger backed himself rather than use the supporting Downey on the inside he was caught. Christian Day was on his shoulder but by then the blue and black shirts were back to cover and the lock was forced into touch.
The visitors nevertheless earned themselves a penalty for their efforts, from which Lamb extended the lead. But inevitably the hitherto one-way traffic was going to abate, and thanks to a pair of Stephen Donald penalties just past the half-hour they found themselves very much back in the match.
But with Lamb landing a smart drop-goal when his team got back to Bath's 22 for the first time in 15 minutes, and then a penalty on the stroke of half-time, it was the visitors who ended the half positively with a 10-point lead.
Bath had showed in the second quarter that they can never be counted out at home, and when Donald failed to reward a Ryan Caldwell break by missing his next shot at goal Saints had received a let off.
Less than 60 seconds later however a Matt Banahan break put Saints massively under pressure. Banahan kicked ahead, Matt Carraro and Tom Biggs gave chase and Ashton put the ball over the dead ball line.
Still the hosts had the five-metre scrum, but with Carraro fumbling a Donald pass under pressure Saints had escaped once more. It got even better when the pack backed themselves at the scrum, winning a penalty and clearing their lines.
But when Mark Sorenson saw yellow for not rolling away the Rec's volume rose noticeably, and while some outstanding defence - including a try saver from Foden on Abendanon - kept the hosts out they were still in a perfect position to pile on the pressure with a five-metre scrum.
However their mistakes again proved their undoing, a Donald fumble adding to two missed penalties to let the visitors off the hook. Even the seven-man Saints scrum had too much power and nous for their opponents, turning on the power to win a penalty.
Having done the hard work Saints then won the 10-minute sin bin period 3-0 thanks to Lamb's third penalty, and things got even better just past the hour. First Guy Mercer saw yellow for not rolling away, then Andy Long found himself at the bottom of a driving maul to score against his former club.
Stephen Myler kept up the fly halves' superb day off the tee by adding the extras and with a quarter of an hour to play Saints found themselves with a 20-point lead.
Bath were now resorting to Sevens-style rugby to try and get back in the game, and while they steadily made metres upfield they found the swarming visitors' defence too tough to break down and eventually turned the ball over with a loose Banahan pass into touch.
It summed Bath's afternoon up, and Saints ground down the clock to record a victory which keeps the push for the play-offs on course.
REFEREE Martin FOX
12mins SAINTS Try Penalty Con Lamb 0-7
25mins SAINTS Pen Lamb 0-10
30mins BATH Miss pen Donald
32mins BATH Pen Donald 3-10
35mins BATH Pen Donald 6-10
38mins SAINTS DG Lamb 6-13
40mins SAINTS Pen Lamb 6-16
HALF-TIME BATH 6 SAINTS 16
42mins BATH Miss pen Donald
47mins SAINTS Yellow card Sorenson
54mins BATH Miss pen Donald
58mins SAINTS Pen Lamb 6-19
61mins BATH Yellow card Mercer
63mins SAINTS Try Long Con Myler 6-26
FULL-TIME BATH 26 SAINTS 6