The Blues went in front on the half-hour mark before an Ethan Ampadu own goal and a Loic Nego volley put the home side in front, only for Giroud to level the game with a superb set piece after replacing the injured Alvaro Morata.
The result sees Chelsea miss out on registering a 100% record in Group L, while Videoton end the group stages in third place with BATE Borisov defeating PAOK.
Chelsea dominated possession during the opening stages of the contest, but the only chance for the Blues during the first quarter-of-an-hour saw Morata glance a header off target from a corner.
Videoton were not allowing Chelsea to have it their own way and created an opportunity of their own in the 24th minute, only for Nego to curl a shot too close to Willy Caballero.
Chelsea were struggling to get anything going in the final third, but on the half-hour mark Willian put his side in front by sending a delightful free kick into the bottom corner from 25 yards.
However, that advantage lasted less than two minutes. An inswinging corner from the right was flicked on at the near post and Ampadu could do little but head the ball into his own net from six yards out.
Chelsea soon went in search of a second goal with Morata being denied from close range, but the Spaniard sustained a knee injury which would force him off before half time.
Videoton also wasted a chance before the break as Caballero pulled off a solid intervention to palm Nego's effort away for a corner.
The home side went onto the offensive at the start of the second half and they should have been ahead when Stopira was found at the back post, but the defender's free header went several yards over the bar.
However, Videoton did go in front minutes later - and in some style, too. A floated pass from left to right found Nego and the defender unleashed a thunderous first-time volley inside Caballero's near post from 15 yards out.
Chelsea quickly went in search of an equaliser with Callum Hudson-Odoi seeing a weak effort saved by Adam Kovacsik and substitute Giroud having a well-struck volley at the back post go whistling over the bar.
Giroud soon found himself on the scoresheet, however. The Frenchman took responsibility for a free kick on the edge of the area and sent a precise effort into the top corner via the underside of the crossbar.
Both sides pushed for a winner during the closing stages of the game, but chances were few and far between as Chelsea missed out on a sixth successive win and Videoton failed to complete what would have been one of the biggest victories in their history.
VIDEOTON (4-3-3): Kovacsik; Fiola, Juhasz, Vinicius, Stopira; Nego, Hadzic (Patkai 85'), Nikolov; Milanov (Kovacs 81'), Scepovic, Huszti (Hodzic 81')
CHELSEA (4-3-3):: Caballero; Zappacosta, Christensen, Ampadu, Palmieri; Loftus-Cheek, Fabregas, Barkley; Willian (Willian 56'), Morata (Giroud 45'), Hudson-Odoi