The Trotters were behind inside six minutes at the Reebok on Saturday as Scott Sinclair found the top corner but it took them only eight minutes to equalise through Chris Eagles.
Swansea laid siege to the Bolton goal in the second half, with Danny Graham squandering a glorious chance by shooting tamely at Adam Bogdan before powering an effort against the bar, while Eagles shot just wide with a late chance for the hosts.
Blackburn's victory over Norwich on Saturday dropped Bolton to 19th but they still have games in hand over their rivals and this draw saw them close to within one point of QPR in 17th.
There was no doubt Bolton needed the points more but the visitors settled quickly and took the lead in the sixth minute with a goal of real quality from Sinclair.
The winger collected the ball on the left and took advantage of Gretar Steinsson standing off him to run along the edge of the box and curl a superb shot into the top corner.
Bolton sought a quick response and they got it when, in the 14th minute, Martin Petrov broke away down the left and pulled the ball back for the onrushing Eagles to tuck away.
The hosts had not learned their lesson from the opener, though, and were still giving Swansea too much time and space around the area.
They almost paid the price when a deflected cross fell to dangerman Gylfi Sigurdsson on the penalty spot but Steinsson did brilliantly to block the shot.
Swansea were forced into an early change when Luke Moore replaced Nathan Dyer in the 29th minute, while Ashley Williams became the first man in referee Phil Dowd's notebook for a foul on Kevin Davies 20 yards out.
It was a great position and Petrov's shot looked to be heading for the bottom corner until Michel Vorm stuck out a hand.
Sigurdsson was then given an opportunity for one of his long-range specials when he was pulled back by Mark Davies 25 yards out. Bogdan could only parry the free-kick and he was fortunate the ball fell to defender Steven Caulker, who miscued his shot.
Swansea certainly looked the more likely scorers and Bogdan was again let off the hook when he came for a 42nd-minute corner and got nowhere near it, with Williams flicking his header well wide.
Swansea were again quickly out of the blocks at the start of the second half and only some last-ditch scrambling from the Bolton defence prevented Williams from squeezing home Sigurdsson's corner.
A lovely through ball from Joe Allen then played in Graham but he poked his shot rather tamely at Bogdan when he really should have scored.
Bolton were hanging on, with Swansea's latest chance seeing Graham pop up on the left and curl a shot just wide of Bogdan's far post.
Coyle sent on David Ngog for Ryo Miyaichi, who had struggled to get into the game, but the onslaught continued and Sinclair would surely have tapped in Graham's cross-shot but for a timely intervention from Steinsson.
Graham, Sinclair and Sigurdsson had run rings around Bolton and they combined again in the 73rd minute, with Sigurdsson playing in Graham for a shot that Bogdan could only parry.
Sinclair tried three times to turn the ball in but was denied by a combination of Tim Ream and Steinsson, while moments later Graham thundered a shot against the bar.
Having survived the siege, Bolton began to put some pressure on the Swansea goal in the final 10 minutes and Eagles came within a whisker of finding the net when his shot from the edge of the box flew just wide with Vorm beaten.
Swansea still looked dangerous and Moore curled a shot just over, but it was not to be for Rodgers' side, while Bolton's last act was to give a Premier League debut to 18-year-old Josh Vela in place of the hobbling Mark Davies.