The visitors, after absorbing initial Leicester pressure, made a dream start following wing Brendon Daniel's powerful run out of defence.
Daniel made 50 metres to just inside Leicester's half, and quickly recycled possession saw centre Robbie Fleck make an incisive break through some weak tackling and send supporting flanker Michael Lipman crashing over for a fifth-minute try that Olly Barkley converted.
Barkley landed a 35-metre penalty on 11 minutes, and the Leicester tempers began to fray, while Bath appeared comfortable and composed in comparison.
Barkley should have extended Bath's lead nine minutes before half-time, but he fluffed an easy penalty chance, giving Leicester an opportunity to at least end the first period by mounting a threatening passage of play.
The Tigers then sacrificed a kickable penalty for an attacking scrum five metres out, yet Bath were in the mood to absorb anything Leicester could throw at them.
After the home side failed to prosper from five set-scrums, Barkley's mammoth 70-metre kick to touch once again gave the visitors territorial control.
Leicester failed to break through the Bath defence, and Barkley ended the half with a 50-metre penalty strike to give Bath a 13-0 lead going into the break.
The hosts finally opened their account in the 46th minute when full-back Geordan Murphy landed an angled penalty, and Broadfoot was replaced at fly-half by Matt Cornwell.
Bath responded by launching an attacking thrust through Fleck – and although Leicester defended resiliently, they suffered an injury blow when their England prop Julian White departed the action holding his left shoulder.
With Gloucester's Phil Vickery having only just started his competitive comeback after back surgery, White's premature exit could prove a cause of concern for caretaker national coach Andy Robinson.
But the Tigers continued to threaten and in the 54th minute No.8 Martin Corry ran aggressively into the heart of Bath's defence, allowing a crisp handling move to create an overlap allowing Murpyh to cross in the corner.
The Irish international could not convert, but Leicester were back in contention – 13-8 adrift approaching the final quarter.
Barkley, however, opened up an eight-point gap when he slotted drop-goal shorlty after, taking the wind out of Leicester's sails after the home side had threatened a revival.
It was impressively efficient rugby from Bath, who began the current league campaign with successive defeats against Northampton and Newcastle but had now moved through the gears following that stuttering start.
Broadfoot, back on the field for an injured Cornwell, drifted a penalty chance wide 11 minutes from time – increasing frustration levels among a capacity 16,815 crowd.
Leicester launched the closing 10 minutes by establishing a foothold inside Bath's 22, and Broadfoot's clever kick towards the posts almost produced a try-scoring chance.
Bath continued to keep their composure in defence before Broadfoot booted a short-range penalty as the clock ticked down – cutting Leicester's deficit to five points.
Leicester's forwards went for broke in the dying minutes, rumbling their way ominously towards the Bath line – and the visitors had to defend grimly in injury-time.
But Bath were unlocked when scrum-half Harry Ellis broke on the blindside of a scrum and touched down in the corner. The video official was called for, and after a lengthy delay the try was awarded.
Broadfoot missed the touchline conversion, which left the score at 16-16 as play continued.
But Bath had let a 13-0 advantage slip away and could only kick themselves as Leicester collected two points for the draw and moved top of the table above Sale Sharks.
For Leicester:Tries: Murphy, EllisPens: Murphy, Broadfoot
For Bath:Try: LipmanCon: BarkleyPens: Barkley 2DG: Barkley