Lions forward Keogh chipped his spot-kick down the middle of the goal after being brought down by Zat Knight in the ninth minute.
But fellow Republic of Ireland international Andrews ensured it was all square at the interval when former Wanderers defender Danny Shittu fouled Lee Chung-Yong.
Both sides had their moments during the second half before Millwall repelled a frenzied late rally from their hosts.
Some slack defensive play from Wanderers gifted Millwall an early lead. Tim Ream's pass into midfield sold Andrews short and Liam Trotter stole in to release Keogh. The former Wolves striker had his heels clipped by Knight and picked himself up to cheekily beat Adam Bogdan.
Lee bustled forward in an attempt to engineer a Bolton response and teed up David Ngog, but the ex-Liverpool man's effort was scuffed into the arms of David Forde.
Kevin Davies made a customary nuisance of himself as Millwall failed to deal with a Chris Eagles free-kick, leaving Lee to fizz wide through a crowded penalty area.
Jackett's men looked to re-assert themselves and Bogdan was not breathing easily when Trotter and James Henry produced superb efforts from distance that were narrowly off target.
Wanderers were level 11 minutes before the break after a deft Davies pass sent Lee into the area and Shittu clumsily intervened. Andrews opted for a more conventional method than Keogh, striking the spot-kick low into the bottom left corner.
Another speculative Millwall effort, this time from Martyn Woolford as the ball dropped kindly onto his left foot on the edge of the area, had Bogdan scrambling once more moments later.
Davies might have done better with a headed attempt in first-half stoppage time but Wanderers took their momentum into the second period.
Stifled appeals for handball against Shittu were waved away when he blocked an Andrews effort and, in the 55th minute, Eagles arrived to plant a volley wide from Davies' knock-down, following a superb run and cross from full-back Sam Ricketts.
After the hour, Ngog led Millwall a merry dance with a jinking run before firing over, and midway through the half the tide turned once more with Henry pivotal for Millwall.
Dany N'Guessan almost turned home his cross from the right before the ex-Reading man drove past Marcos Alonso and Lee and saw the resulting shot spin behind off Ream.
Forde produced the save of the match in the 83rd minute, turning Alonso's 25-yard drive behind despite being partially unsighted.
Davies muscled his way onto the resulting Jay Spearing corner but his header was hacked off the line by N'Guessan and Millwall scrambled clear and in a flurry of late Bolton pressure, substitute Marvin Sordell latched on to Davies' flick-on but flashed just wide.